STABLE
NON-GAUSSIAN
RANDOM
PROCESSES

STOCHASTIC MODULI \(;
Series Editors
Laurence Baxter
State University of New York at Stony Brook
Marco Scarsini
Universita D'Annunzio
Moshe Shaked
University of Arizona
Stadler Stidham, Jr.
University of North Carolina
G. Samorodnitsky and M.S. Taqqu
Stable Non-Gaussian Processes:
Stochastic Models with Infinite Variance
K. Sigman
Stationary Marked Point Processes: An Intuitive Approach
•*■'*$: V*
*%*>.

STABLE
NON-GAUSSIAN
RANDOM
PROCESSES
Stochastic Models
with Infinite Variance
STOCHASTIC MODELING
Gennady Samorodnitsky
Murad S. Taqqu
CHAPMAN & HALL/CRC
Boca Raton London New York Washington, D.C.

Library of Congress has Cataloged the Hard Covered Imprint Edition as Follows:
Samorodnitsky, Gennady.
Stable non-Gaussian random processes : stochastic models with infinite
variance / Gennady Samorodnitsky, Murad S. Taqqu.
p. cm.
Includes bibliographical references and indexes.
ISBN 0-412-05171-0
1. Gaussian processes. 2. Gaussian distribution.
I. Taqqu, Murad S. II. Title.
QA274.4.S26 1994
519.2—dc20 94-13685
CIP
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Printed on acid-free paper

To Julia, Eric and Danny
and
to Rachelle, Yael and Jonathan

Contents
Preface xiii
Abbreviations xvii
Notation xix
1 Stable random variables on the real line 1
1.1 Equivalent definitions of a stable distribution 2
1.2 Properties of stable random variables 10
1.3 Symmetric a-stable random variables . 20
1.4 Series representation 21
1.5 Series representation of skewed Q-stable random variables ... 30
1.6 Graphs and tables of a-stable densities and c.d.f.'s 35
1.7 Simulation 41
1.8 Exercises 49
2 Multivariate stable distributions 55
2.1 Stable random vectors 57
2.2 A counterexample for 0 < a < 1 63
2.3 Characteristic function of an a-stable random vector 65
2.4 Strictly stable and symmetric stable random vectors 72
2.5 Sub-Gaussian random vectors 77
2.6 Complex SaS random variables 84
2.7 Covariation 87
2.8 Covariation norm 95
2.9 James orthogonality 97
2.10 Codifference 103
2.11 Exercises 107

Vlll CONTENTS
3 Stable random processes and stochastic integrals 111
3.1 Stable stochastic processes 112
3.2 Definition of stable integrals as a stochastic process 113
3.3 Qrstable random measures 118
3.4 Constructive definition of stable integrals 121
3.5 Properties of stable integrals 126
3.6 Examples 135
The SaS Levy motion 135
Moving averages 138
Omstein-Uhlenbeck process 138
Reverse Omstein-Uhlenbeck process 139
Well-balanced linear fractional stable motion 140
Log-fractional stable motion 141
Real stationary SaS harmonizable process 141
3.7 Sub-Gaussian processes 142
3.8 Sub-stable processes 143
3.9 Series representation for a-stable random measures 145
3.10 A third definition of stable stochastic integrals using the series
representation 149
3.11 Conditions 152
3.12 A fourth definition of stable stochastic integrals using a Poisson
representation 155
3.13 Exercises 167
4 Dependence structures of multivariate stable distributions 173
4.1 Linear regression 174
4.2 Conditional laws that are symmetric around the conditional mean 181
4.3 Linear dependence 185
4.4 Probability tails of order statistics 187
4.5 Joint moments 200
4.6 Association of stable random variables 204
4.7 The codifference for stationary SaS processes 208
4.8 The expected number of level crossings for stationary sub-
Gaussian processes 215
4.9 Exercises 217
5 Non-linear regression 223
5.1 Conditional moments of order greater than or equal to a .... 224
5.2 Analytic representations of the non-linear regression functions . 236
5.3 Examples 251

CONTENTS
IX
5.4 Graphical representations 255
5.5 Numerical techniques 260
5.6 Exercises 270
6 Complex stable stochastic integrals and harmonizable processes 271
6.1 Complex-valued SaS random measures 272
6.2 Integrals with respect to complex-valued SaS random measures 275
6.3 The complex isotropic SaS case 281
6.4 Series representation of complex-valued SaS random measures
and integrals 286
6.5 Harmonizable process 291
6.6 Stationary real harmonizable processes 300
6.7 The codifference for stationary real harmonizable processes . . 305
6.8 Exercises 306
7 Self-similar processes 309
7.1 Self-similarity 311
7.2 Fractional Brownian motion 318
7.2.1 "Moving average" representations of fractional Brownian
motion 320
7.2.2 "Harmonizable" representations of fractional Brownian
motion 325
7.2.3 Fractional Gaussian noise 332
7.3 General characteristics of processes that are a-stable and iJ-sssi 340
7.4 Linear fractional stable motion 343
7.5 a-stable LeVy motion 349
7.6 Log-fractional stable motion 352
7.7 The real harmonizable fractional stable motion 355
7.8 Complex harmonizable fractional stable motion 358
7.9 Subordinated processes 363
7.10 Fractional stable noises 366
7.11 Simulation of fractional noises and motions 370
7.12 ARMA sequences with stable innovations 376
7.13 Fractional ARIMA with stable innovations 380
7.14 Exercises 387
8 Chentsov random fields 391
8.1 Self-similar fields with stationary increments in the strong sense 392
8.2 Chentsov random fields 394
8.3 Example: the L6vy-Chentsov random field 400

X
CONTENTS
8.4 Example: Takenaka random fields 402
8.5 Properties of Chentsov random fields 405
8.6 Properties of H-sssis Chentsov random fields 407
8.7 Codifference induced by (a, iJ)-Takenaka fields 410
8.8 Takenaka processes on [0, oo) 414
8.9 Exercises 417
9 Introduction to sample path properties 419
9.1 Versions 420
9.2 Separability 421
9.3 Applications 427
9.4 Measurability 430
9.5 Zero-one laws 434
9.6 Exercises 439
10 Boundedness, continuity and oscillations 445
10.1 Introduction 446
10.2 Necessary conditions for sample boundedness 447
10.3 Necessary conditions for sample continuity 455
10.4 Necessary and sufficient conditions for sample boundedness and
continuity when 0 < a < 1 460
10.5 Probability tails of suprema of bounded a-stable processes, with
index 0 < a < 2 470
10.6 The oscillation process 476
10.7 The case 0 < a < 1 482
10.8 The case 1 < a < 2 483
10.9 The level sets of the oscillation function 484
10.10 A sample path alternative 486
10.11 How strong is the basic assumption? 488
10.12 Exercises 490
11 Measurability, integrability and absolute continuity 497
11.1 Existence of a measurable version 498
11.2 Integrability of the sample paths of stable processes 502
11.3 Conditions for integrability 504
11.4 Changing the order of integration . . . 511
11.5 Tail behavior of the Lp-norm distribution 515
11.6 Inversion formula for harmonizable SaS processes 519
11.7 Absolute continuity of stable processes 524
11.8 Exercises 533

CONTENTS xi
12 Boundedness and continuity via metric entropy 537
12.1 Metric entropy 538
12.2 Sufficient conditions in the case 1 < a < 2 542
12.3 Necessary conditions in the case 1 < a < 2 546
12.4 Boundedness and continuity of self-similar a-stable processes . 550
12.5 Exercises 556
13 Integral representation 559
13.1 Countable parameter space 560
13.2 Arbitrary parameter space 568
14 Historical notes and extensions 571
14.1 Notes to Chapter 1 571
14.2 Notes to Chapter 2 575
14.3 Notes to Chapter 3 577
14.4 Notes to Chapter 4 578
14.5 Notes to Chapter 5 582
14.6 Notes to Chapter 6 585
14.7 Notes to Chapter 7 586
14.8 Notes to Chapter 8 590
14.9 Notes to Chapter 9 592
14.10 Notes to Chapter 10 592
14.11 Notes to Chapter 11 593
14.12 Notes to Chapter 12 594
14.13 Notes to Chapter 13 595
Appendix: 597
A Tables of symmetric a-stable fractiles 597
Bibliography 603
Subject index 621
Author index
629

Preface
This book has been written to fill a gap that we, as teachers and researchers in the
field in probability, have increasingly felt. Stable processes, which have attracted
growing interest in recent years, are not the single subject of any monograph or
comprehensive overview. In this book, we hope to make this important branch
of probability widely accessible and provide both an introduction and a basic
reference text.
The central limit theorem which offers the fundamental justification for
approximate normality points to the importance of the stable distributions: they
are the only limiting distributions of normalized sums of independent, identically
distributed random variables, and perforce include the Gaussian as distinguished
elements. Gaussian distributions and processes have long been well understood
and their utility as both stochastic modeling constructs and analytical tools is
well-accepted. However, they do not allow for large fluctuations and are thus
often inadequate for modeling high variability. Non-Gaussian stable models, on
the other hand, do not share such limitations. In general, the upper and lower tails
of their marginal distributions decrease like a power function. The rate of decay
depends on a number a, which takes a value between 0 and 2. The smaller a, the
slower the decay and the heavier the tails. The distributions always have infinite
variance and when a < 1, they have an infinite mean as well.
In the last two or three decades, data with "heavy tails" have been collected
in fields as diverse as economics, telecommunications, hydrology and physics of
condensed matter, which suggests using non-Gaussian stable processes as possible
models. Such models offer the additional merit of flexibility and variety when
compared to Gaussian processes. The latter are completely specified by their mean
and autocovariance functions, whereas non-Gaussian stable processes command
a much richer parameterization. Gaussian distributions, moreover, are always
symmetric around their mean; the non-Gaussian stable ones can have an arbitrary
degree of skewness.
In this book, we emphasize the probabilistic approach over the analytic one.
We talk of tails, moments and dependence structures and focus on multivariate

XIV
PREFACE
properties and sample paths. The book will be useful to a wide spectrum of
researchers in probability, applied probability and statistics as well as to graduate
students. As background, we require only a first year graduate course in
probability. Our goal has been to write a very readable text and to keep the wider context
in clear perspective. Proofs are presented in detail. Each chapter begins with a
brief summary and concludes with a wide range of exercises at varying levels of
difficulty. To guide students on the more challenging problems, we have supplied
detailed hints.
In Chapter 1 we introduce the one-dimensional stable random variables and
provide a foundation for the subsequent chapters. We review classical material
from the first part of this century, and then present more recent developments,
including sections on simulation and on the series representation. That
representation, which provides a link between the classical theory and the new, more
modem approaches, is used extensively in the subsequent chapters.
Much of the rich structure of the stable non-Gaussian world becomes already
apparent in Chapter 2 where multivariate stable distributions are presented and
fundamental notions like spectral measure, covariation, and codifference are
introduced. While the reader may notice that some of the Gaussian tools remain
useful, this chapter is more a story of contrast than of similarity between Gaussian
and non-Gaussian stable multivariate distributions. One important difference
is that the components of a stable non-Gaussian vector cannot, in general, be
expressed as a finite linear combination of independent random variables. Instead,
one has to use stable stochastic integrals as a representation. These have enough
built-in "independence" and are sufficiently versatile to represent non-Gaussian
stable processes. Stable stochastic integrals are defined in several ways in Chapter
3, where the reader first encounters some of the most important classes of stable
processes.
In Chapter 4 we study in greater detail the dependence structure of multivariate
stable distributions: the effect of conditioning, order statistics, existence of joint
moments, association and codifference are some of the topics that are discussed.
Simple regression is analyzed in depth in Chapter 5. While non-Gaussian stable
distributions have only moments of order less than a, they may have conditional
moments of order greater than a. In particular, a regression of one symmetric
a-stable random variable on another may be well defined even with a < 1; it is
then linear as in the Gaussian case. When the stable distributions involved are
no longer symmetric, the situation is more complicated and the regression, in this
case, is typically non-linear. We provide exact forms for the regression.
Complex-valued symmetric stable random variables, briefly introduced in
Chapter 2, are discussed in greater detail in Chapter 6. In that chapter we develop
the theory of complex-valued stable integrals and stochastic processes, together

PREFACE
XV
with their series representation, and introduce the harmonizable stable processes.
Harmonizable or "spectral" representations are commonly used in Gaussian
processes; in the non-Gaussian stable context, they characterize a special class of
stochastic processes.
In Chapter 7 we discuss self-similar processes, also called "random fractals".
These processes are invariant in distribution under judicious scaling of time and
space. They are important in probability because of their connection to limit
theorems and they are of great interest in modeling because they can display
"long-range dependence" or "long memory." Some aspects of self-similarity
appear in geophysics, hydrology, turbulence, economics, communications and in
relation to "l/f noises." Self-similar processes are also used in physics,
particularly in connection to the so-called "renormalization group theory" and "critical
phenomena." We start by introducing the Gaussian ones, namely the fractional
Brownian motions, and then concentrate on the much richer family of stable non-
Gaussian self-similar processes. We also discuss autoregressive moving-average
(ARMA) and fractional ARIMA models with stable innovations. The latter can
display high variability together with long-range dependence.
In Chapter 8 we consider Chentsov random fields. We show how one can
extend Chentsov's geometric construction of L6vy Brownian motion to construct
a variety of self-similar stable random fields.
In the following four chapters we consider the sample path properties of stable
processes. Chapter 9 is of an introductory nature. Its purpose is to help a reader
with little prior experience to gain familiarity with the basic notions in the field,
such as versions, separability, zero-one laws, etc. As a result, much of the
discussion in Chapter 9 is not specific to stable processes but is applicable to any
stochastic process. An advanced reader may wish to skip the first three sections
of that chapter.
In Chapter 10 we discuss the most basic sample path properties of stable
processes, namely sample boundedness and continuity. Some stable processes are
continuous, others have discontinuities of the first type, and there are some that are
unbounded in any finite interval. Self-similar stable processes provide a number
of examples of these different types of sample paths. We also describe the upper
tails of the distribution of suprema of stable processes and include a discussion on
oscillations. In Chapter 11 we consider the measurability and integrability of the
sample paths. One application is a formula for the inversion of the harmonizable
process which is similar to the one for ordinary Fourier transforms.
The description of sample boundedness and continuity of stable processes
given in Chapter 10 is complete only when 0 < a < 1. In Chapter 12 we offer a
different approach for the case 1 < a < 2, one based on metric entropy. Although
Chapter 12 does not contain a full solution to the problem, it provides new insight

XVi PREFACE
into boundedness and continuity in the stable case.
A promise made to the reader in Chapter 3 is kept in Chapter 13, where
a detailed discussion of integral representation of a-stable processes is given.
Finally, Chapter 14 contains historical notes, references and extensions to the
material presented in the text.
There are many ways to read this book. Chapters 1,2 and 3 are prerequisites for
all the subsequent ones. Together with Chapter 4, they provide a good overview of
the subject. The remaining chapters contain more specialized material and can be
read independently. Chapters 9 through 12 which cover the sample paths' behavior
have a more abstract flavor. Chapter 13 on integral representation requires only
Chapter 3 as background. In a one-semester course one can cover the introductory
chapters, 1 through 4, as well as Chapter 7 on self-similar processes.
We are grateful to the Weizmann Institute and the Guggenheim Foundation.
The first offered a fellowship to Gennady Samorodnitsky, the second to Murad
Taqqu. These fellowships allowed us to start the project. The National Science
Foundation, the Air Force Office of Scientific Research and the Office of Naval
Research provided us with grants to investigate stable processes. We thank also
Albert Paulson for allowing us to reproduce some stable distribution tables from
an unpublished technical report and the following publishers for their kind
permission to reproduce some material from their publications: Birkhauser, Blackwell
Publishers, Academic Press, Elsevier Press, the American Statistical Association
and the Institute for Mathematical Statistics.
Many people read parts of the manuscript and provided helpful suggestions.
We would like to thank Laurence Baxter, Stamatis Cambanis, Daniel Chambers,
Jan Kallsen, Mark Kon, Michel Ledoux, Makoto Maejima, Joop Mijnheer, Jolanta
Misiewicz, Svetlozar (Zari) Rachev, Balram Rajput, Sid Resnick, Yumiko Sato,
Shigeo Takenaka, Wim Vervaat and Tomasz Zak. We are also indebted to our
students Renata Cioczek-Georges and Piotr Kokoszka, who were closely involved
with many substantive aspects of this project, and Vadim Teverovsky who designed
most of the figures. Alex Kasman, supported by the NSF grant USE-8953023,
developed "calcgraphics," which allowed us to combine Mathematica and IATgX.
At the same time, Ognian Enchev showed us how to work around some of IATgX's
idiosyncrasies. We had the good fortune to have Tom Orowan, a superb typist
and friend, type this manuscript at Boston University. And finally, we thank our
wives, Julia and Rachelle, for their support and understanding.

Abbreviations
a.s. almost surely
ARIMA autoregressive integrated moving average process
ARMA autoregressive moving average process
CHFSM complex harmonizable fractional stable motion
FARIMA fractional autoregressive integrated moving average process
FBM fractional Brownian motion
FGN fractional Gaussian noise
H-ss self-similar with index H
H-sssi self-similar with index H and with stationary increments
F-sssis self-similar with index H and with stationary increments
in the strong sense
LFSM linear fractional stable motion
log-FSM log-fractional stable motion
RHFSM real harmonizable fractional stable motion
SaS symmetric a-stable

Notation
= equality in distribution
1,4 or 1 (A) indicator function of the set A
a<p> signed power, equal to |a|psign a
ax ~ bx lim ax/bx = 1
A—*oo
a\ « 6a lim a\/b\ — k for some k ^ 0
A—»oo
6(xo) or 6{I0} measure that gives unit mass to the point xq
plim limit in probability
Leb Lebesgue measure
N(/j,, v1) normal distribution with mean /z and variance v2
sign(x) lifrr > 0, 0ifx=0, -lifx<0
Sa (a, P, ix) stable distribution with index a, scale parameter a,
skewness parameter P and shift parameter /i; in
particular, ^(ct,0, fj.) = N(n,2a2)
Sd unit sphere in M.d ; it is a (d - 1)-dimensional surface
X ~ Sa (<?, P, m) X has the distribution SQ (a, P, /x)
|| X || Q scale parameter of a SaS random variable X
x+ x if x > 0, 0 if x < 0

STABLE
NON-GAUSSIAN
RANDOM
PROCESSES

Chapter 1
Stable random variables on
the real line
This chapter concerns univariate stable distributions. Because these distributions
are described in several textbooks and monographs, we do not include the proof
of some well known results. Instead, we refer the reader to Feller (1971) or
Gnedenko and Kolmogorov (1954) where such proofs can be found. Our goal is
to provide a clear summary of the classical theory. Modem developments, such
as the series representation, are presented in detail.
We define univariate stable distributions in four equivalent ways. The first
two definitions concern the "stability" property: the family of stable distributions
is preserved under convolution. The third concerns the role of stable distributions
in the context of the central limit theorem. Stable distributions approximate the
distribution of normalized sums of i.i.d. random variables making them useful in
modeling the contribution of many small random effects. The fourth definition
specifies the characteristic function of a stable random variable. We will use the
characteristic function extensively because few stable density functions are known
in closed form.
A univariate stable distribution is characterized by four parameters. These are
the index of stability a, the scale parameter a, the skewness parameter /? and the
shift parameter \i. The stable distribution is Gaussian when a = 2, and in this
case, a is proportional to the standard deviation, /? can be taken to be zero and \i
is the mean.
In Section 1.2 we present some basic properties of stable distributions. We
clarify the role played by the four parameters. Stable distributions with a < 2
share many properties with the Gaussian distribution, but they also differ from
the Gaussian in significant ways. When a < 2, for example, the tails of the

2 ' STABLE RANDOM VARIABLES ON THE REAL LINE 1.1
distributions decay like a power function. This means that a stable random
variable exhibits much more variability than a Gaussian one: it is much more
likely to take values far away from the median. Mandelbrot (1982) referred to this
as "Noah effect," an allusion to the biblical figure who lived through a very severe
flood. The high variability of the stable distributions is one of the reasons they
play an important role in modeling. Stable distributions have been used to model
such diverse phenomena as gravitational fields of stars, temperature distributions
in nuclear reactors, stresses in crystalline lattices, stock market prices and annual
rainfall.
In Section 1.3, we discuss symmetric stable distributions, which are a-stable
with /? and /x zero. Their characteristic function takes a particularly simple form,
very similar to that of a centered Gaussian distribution.
We establish in Sections 1.4 and 1.5 the series representation for a stable
random variable. We show that any stable random variable can be expressed
as an infinite sum which involves the arrival times of a Poisson process. This
representation clarifies the role played by the index of stability a. Analogous
representations will be established for stable random vectors and stable stochastic
processes.
In Section 1.6, we display graphs of univariate stable density and cumulative
distribution functions and provide a guide to numerical tables. We show, in Section
1.7, how to simulate stable random variables.
1.1 Equivalent definitions of a stable distribution
The theory of univariate stable distributions was essentially developed in the 1920s
and 1930s by Paul L6vy and Aleksander Yakovlevich Khinchine. It is covered
in detail in such classics as Gnedenko and Kolmogorov (1954) and Feller (1971),
and it is the object of a more recent monograph of Zolotarev (1986). The main
features of the theory are outlined in many graduate textbooks in probability.
We give four equivalent definitions of a stable distribution.
Definition 1.1.1 A random variable X is said to have a stable distribution if for
any positive numbers A and B, there is a positive number C and a real number D
such that
AXt + BX2 = CX + D, (1.1.1)
where X\ and X2 are independent copies of X, and where " =" denotes equality
in distribution.

1.1 EQUIVALENT DEFINITIONS OF A STABLE DISTRIBUTION 3
Note that a random variable X concentrated at one point is always stable. This
degenerate case is of no special interest and, unless stated explicitly, we always
assume that X is non-degenerate. A random variable X is called strictly stable if
(1.1.1) holds with D = 0. A stable random variable X is called symmetric stable
if its distribution is symmetric, that is, if X and —X have the same distribution.
A symmetric stable random variable is obviously strictly stable.
Warning. In the older literature, e.g. L<Svy (1954) and Feller (1966), the terms
"stable" and "strictly stable" are replaced, respectively, by "quasi-stable" and
"stable." Feller (1971) uses, respectively, "stable (in the broad sense)" and "stable
(in the strict sense)."
Theorem 1.1.2 For any stable random variable X, there is a number a € (0,2]
such that the number C in (1.1.1) satisfies
Ca = Aa + Ba. (1.1.2)
See Feller (1971), Section VI. 1, for a proof. The number a is called the index of
stability or characteristic exponent. A stable random variable X with index a is
called a-stable.
Example 1.1.3 If X is a Gaussian random variable with mean n and variance v2
(X ~ N(n, i/2)), then X is stable with a = 2 because
AXi + BX2 ~ N{{A + B)n, {A2 + B2)u2),
i.e., (1.1.1) holds with C = {A1 + B2)^'1 and D = (A + B - C)n-
We now turn to the second definition.
Definition 1.1.4 {equivalent to Definition 1.1.1). A random variable X is said to
have a stable distribution if for any n > 2, there is a positive number C„ and a
real number Dn such that
Xx+X2 + --- + Xn = CnX + Dn, (1.1.3)
where X\,X2,- ■ ■, Xn are independent copies of X.
If X is stable according to the Definition 1.1.1, then, by induction, it is
also stable according to Definition 1.1.4. It is easy to show that the reverse
implication is also true (Feller 1971, Section VI.l). Therefore the two definitions
are equivalent.
It turns out (Feller 1971, Theorem VI. 1.1), that in (1.1.3) we have, necessarily,
Cn=n'/Q (1.1.4)
for some 0 < a < 2. This is of course the same a which appears in (1.1.2).

4' STABLE RANDOM VARIABLES ON THE REAL LINE 1.1
Remarks
1. If we want to use Definition 1.1.4, is it necessary to verify that (1.1.3) holds
for every w > 2 ? Would it be sufficient to verify (1.1.3) for fewer ns?
It turns out that the requirement Xx + X2 = C2X + D2 alone is not
sufficient, but a random variable satisfying both X\ + X2 = C2X + D2
and X\ + X2 + X3 = dX + D3 is necessarily stable (Zolotarev (1986),
P- 14).
2. Consider a sequence X = {Xi}?l_00 of random variables. Fix a number
6 > 0. For each n > 1, define the transformation Tn that takes X into the
new sequence of random variables TnX = {(TnX)i}fl_00 given by
(i+l)n-l
j=in
To visualize the action of Tn, divide the real line in blocks of size n. In
each block, sum the random variables whose index belongs to the block and
renormalize that sum by a power function of the length of the block. The
index n of Tn refers to the length of the block and 6 is the exponent of the
power function.
The transformations Tn,n > 1, are called renormalization group
transformations with critical exponent 6. More general renormalization group
transformations occur in physics in the context of critical phenomena, and
in applied mathematics.
The family of transformations {Tn,n > 1} forms a semi-group. Indeed,
Tmn = TmTn, i.e., the transformation Tn followed by the transformation
Tm is equivalent to the transformation Tmn. A sequence X = {-X"i}S-oo
is said to be a fixed point of the renormalization group transformation if
TnX = X for all n > 1, i.e., if the distribution of X is invariant under T„
for any n > 1.
The sequence X = {Xi}'?l_00 of i.i.d. strictly a-stable random variables
is a fixed point of the T„s with 6 = \/a because
(TnX)j = ~jj^(Xin + Xin+i H h Xj(n+i)_i) = Xi,
by (1.1.3), and because the independence of the A^s implies the
independence of the (TnX)iS.
We will see in Chapter 7 that there exist dependent stable sequences which
are also invariant under renormalization group transformations. The
corresponding critical exponent 6 will then depend not only on the index of

1.1 EQUIVALENT DEFINITIONS OF A STABLE DISTRIBUTION 5
stability a but also on an additional parameter associated with the
dependence.
The third definition states that stable distributions are the only distributions
that can be obtained as limits of normalized sums of i.Ld. random variables.
Definition 1.1.5 (equivalent to Definitions 1.1.1 and 1.1.4). A random variable
X is said to have a stable distribution if it has a domain of attraction, i.e., if
there is a sequence of i.i.d. random variables Y\,Yi,... and sequences of positive
numbers {dn} and real numbers {an}, such that
Yi+Y2-\ h Yn d ,. ,
han=>A. (.1.1-5)
The notation =>■ denotes convergence in distribution.
It is clear that the previous definitions yield (1.1.5), e.g., by taking the Yis to be
independent and distributed like X. The converse is easy to show (Gnedenko &
Kolmogorov 1954, p. 162). When X is Gaussian and the V^s are i.i.d. with finite
variance, then (1.1.5) is the statement of the ordinary central limit theorem. The
y,s are said to belong to the normal domain of attraction of X when dn = n1/,Q.
In general, dn = nl/ah(n) where h(x), x > .0, is a slowly varying function at
infinity, that is, lim h(ux)/h(x) = 1 for all u > 0 (Feller 1971, XVII.5). The
function h(x) = In x, for example, is slowly varying at infinity.
The fourth definition specifies the characteristic function of a stable random
variable.
Definition 1.1.6 (equivalent to Definitions 1.1.1, 1.1.4 and 1.1.5). A random
variable X is said to have a stable distribution if there are parameters 0 < a <
2, a > 0, -1 < P < 1, and /i real such that its characteristic function has the
following form:
exp^-<7a|6>|Q(l -z0(sign0)tan^)+i/x0V ifo^l,
E&xpiOX = <
expl -cr\9\(\ +i/3l(signfl)ln|0|) +%pti \ if a= 1.
(1.1.6)
The parameter a is the index of stability and
[ 1 if0>O,
sign0= { 0 if 0 = 0,
( -1 if0<O.
The parameters a, /3 and /i are unique (0 is irrelevant when a = 2).

6 STABLE RANDOM VARIABLES ON THE REAL LINE 1.1
It is easy to check that Definition 1.1.6 implies Definition 1.1.4. The proof of
the converse can be found in many textbooks, e.g. Gnedenko and Kolmogorov
(1954), Section 34. The idea of the proof is to use Definition 1.1.5 to obtain the
Livy-Khinchine representation of the characteristic function, namely
EexpiOX = e,xp{iM9 - a292}
if a = 2, and
(1.1.7)
if a < 2. Here M is real, a > 0, P and Q are non-negative numbers, and
tf(0|iC) = e**-l--i^. (1.1.8)
l + x'-
Then, in the case a < 2, when X is non-degenerate (P + Q > 0), one sets
P P + Q
and evaluates the integrals as in Feller (1966), p. 542.
There is a great deal of flexibility in the choice of the function tp. Another
choice, which involves the unit ball [— 1,1] of R1, is
ip(e, x) = ei6x - 1 - i&cl[_Mj(a:).
(Changing i/> affects the values of the constants M, P, Q.)
The measure
P Q
L(dx) = ^iTZho,oo)(x)dx + puj^l^oo^aOds
in (1.1.7), is called the Levy measure.1 We will see in Section 3.12 that the
representation (1.1.7) has an interpretation in terms of limits of compound Poisson
random variables with L(dx) playing the role of the intensity of "jumps" of size
x.
1 L(dx) is sometimes called "spectral measure" in the literature. In order to avoid any possible
confusion with the "spectral measure" of Chapter 2, we refer to L(dx) as the "Levy measure."

1.1 EQUIVALENT DEFINITIONS OF A STABLE DISTRIBUTION 7
Remarks
1. When a = 2, the characteristic function (1.1.6) becomes EtxpiBX =
exp{—a282+ifj.6}. This is the characteristic function of a Gaussian random
variable with mean \i and variance 2a2. Note that a is not equal to the
standard deviation. Note also that although the value of/3 is not specified
because (3 tan n — 0, one typically associates the Gaussian distribution with
the choice 0 — 0.
2. When a = 1, the imaginary term in (1.1.6) contains the factor ln|0|. The
presence of this logarithm is the source of many difficulties associated with
the case a = 1. In the sequel, the case a = 1 will often have to be treated
separately.
3. In (1.1.6), P appears with a negative sign when a ^ 1 and with a positive
sign with a = 1. This minor point has been the source of great confusion
in the literature (see Hall (1980) for a discussion).
4. The characteristic function (1.1.6) can be written as
EexpiOX = exp{aa(-\6\a + idu(6,a,0)) + i^6},
where
W(0,a,/3) = | _0llnm ifa=1>
The function oj(9, a, p) is not continuous at a = 1 and P ^ 0. However,
as Zolotarev (1986) remarks, setting
■ f „ + /?*• tan? if a ,4 1, (u>9)
^ \ fj. if a = 1,
yields the expression
Eexvi6X = exp{(Ta(-\e\a+ieujl{9,a,P)) + ifnd}, (1.1.10)
where
uM'a>® = \-f3lln\e\ ifa=l, (LL11)
is a function which is jointly continuous in a and p. This last expression
makes plausible the presence of the logarithm in the case a = 1.
If one insists on having convergence in distribution as a —► ocq, a —>
ao, (i —* f3o, fi —► po, then one should change the parameterization and

8
STABLE RANDOM VARIABLES ON THE REAL LINE
1.1
characterize an a-stable random variable as having a characteristic function
of the form
Eexpi6X = exp{aa(-\9\a + i9/3(\e\a-i - l)tan^) +tAnfl},
where a 6 (0,2], u > 0, |/?| < 1, iix e R1, and where the characteristic
function for a = 1 is defined by letting a —> 1. The drawback is that /ii
does not have the nice properties of fi given in the next section. We shall,
therefore, continue to use the form (1.1.6) of the characteristic function.
5. When a ^ 1, the characteristic function (1.1.6) is sometimes written as
Eeiex = exp{-tr?|0|<Vw;(eHie"}, (1.1.12)
where
wj(0) = un{B,a,ff) = -ft (sign0)|tf(a),
and where 02 is such that
tan (ft——) =/3tan—,
and K(a) will be defined below. The new scale parameter 02 is related to
a as follows:
02 = CC2,
where
c2=(cos/32|K(a))1/a,
or, in terms of /3,
c2=(l + /32tan2T)
Zolotarev (1986), p. 12, chooses2
{a if q < 1,
a-2 ifa>l.
2Some authors, e. g., DuMouchel (1971) and Chambers, Mallows, and Stuck (1976), choose
{a if a < 1,
2-a if a > 1.
With this choice, fh. has the same sign as P if a < 1, but has the opposite sign if a > 1.

1.1 EQUIVALENT DEFINITIONS OF A STABLE DISTRIBUTION 9
With this choice,
Pi
^Arctan(/3tan^)
if 0 < a < 1,
^r^y Arctan(/3 tan ^~1) if 1< a < 2.
For fixed a, the index fo is a monotone increasing function of /3 and
fo = -1, 0, 1 if and only if/3 = — 1, 0, 1, respectively.
a = 0.9 l
a = 0.5
Figure 1.1: ft versus (3. If a > 1, replace the
a in the figure by 2 — a.
Notation. Since (1.1.6) is characterized by four parameters
c«6(0,2], cr>0, /3 e [-1,1], M£R\
we will denote stable distributions by Sa(a, /?, /i) and write
X ~ Sa{cr,P,fi)
to indicate that X has the stable distribution Sa{cr, j3, /i). We also write
X ~SaS
when X is symmetric a-stable, i.e., as shown in Property 1.2.5 below, when
0 = li = O.
The probability densities of a-stable random variables exist and are continuous
but, with a few exceptions, they are not known in closed form (Zolotarev 1986).
The exceptions are:

10
STABLE RANDOM VARIABLES ON THE REAL LINE
1.2
(a) The Gaussian distribution S2(a, 0, /i) = N(n, 2a2), whose density is
(b) The Cauchy distribution Si (a, 0, fi), whose density is
-77 <T,1 , „■ (1.1.13)
If X ~ S, (ct, 0,0), then for a; > 0,
P(X<x) = i + i-Arctan(-). (1.1.14)
(See Exercise 1.7.)
(c) The Levy distribution Sx/2{a, 1, aO, whose density
(^)I/2(^75-p{-2(^)} (U-15>
is concentrated on (fi, oo). If X ~ 5,1/2(cr, 1,0), then for a; > 0,
P(X<x) = 2(l-<D(y/|)), (1.1.16)
where $ denotes the cumulative distribution function of the AT(0,1)
distribution. (See Exercise 1.9.)
(d) A constant \i which has the degenerate distribution SQ(0,0, \i) for any
0 < a < 2. Our convention is to exclude degenerate distributions because
they have unusual properties. For example, all moments of a degenerate
distribution are finite, whereas a non-degenerate a-stable distribution with
0 < a < 2 has infinite second moments (Property 1.2.16).
1.2 Properties of stable random variables
A useful tool for studying a-stable distributions is the characteristic function
(1.1.6). We shall use it to derive some basic properties of stable random variables
and to obtain an interpretation of the parameters a, <r, /3 and fi.
Property 1.2.1 Let X\ and X2 be independent random variables with Xi ~
Sciendum), i = 1,2. Then X, 4- X2 ~ Sa{a,p,n), with

1.2
PROPERTIES OF STABLE RANDOM VARIABLES
11
PROOF: We verify this for a ^ 1. By independence,
lnEsxpie(Xi + X2) = ln(.Eexpi0Xi) + \a(EexpiBX2)
= -K + *?)l*|a + i\9\a sign (0) (tan ™) (/?,< + /V?) + #(/*, + Pi)
= -(af+a?)\6\"
1 - » J,.„a S1g" *) t3n T"
err + <r" 2
+ ^(jUi+M2)-
The proof for a = 1 is similar. I
The parameter p is a shift parameter because of
Property 1.2.2 Let X ~ Sa(a, P, p) and let a be a real constant. Then X + a ~
5a(o-,/3,/i + a).
This follows trivially from the form of the characteristic function (1.1.6).
Property 1.2.3 Let X ~ Sa(a, P, p) and let a be a non-zero real constant. Then
aX ~ Sa(\a\o, sign (a)P, ap) if a ^ 1 (i o \)
aJC ~ 5i(|a|cr, sign (a)/?, ap — fa(ln |a|)cr/3) </ a = 1.
Proof: By (1.1.6), we have for a ^ 1,
ln{Ee\pi9(aX)}
(7TQ\
1 - iP sign (ad) tan — J + ip(6a)
(7TCKX
1 - iP sign (a) sign (6) tan —J + t(jia)0.
The proof for a = 1 is similar. I
The parameter a is called the scale parameter. Observe that when a = 1,
multiplication by a constant affects the shift parameter in a non-linear way. Hence,
the name "scale parameter" for cr is a misnomer when a — 1 and p =fi 0. When
p = 0, we have
Property 1.2.4 For any 0 < a <2,
X~Sa(<r,P,0) ^=> -X~Sa(a,-p,0). ^22^
The following property identifies /3 as a skewness parameter.
Property 1.2.5 X ~ Sa(o, P, p.) is symmetric if and only ifP-0 and p = 0. It
is symmetric about p if and only ifP = 0.

12 STABLE RANDOM VARIABLES ON THE REAL LINE 1.2
PROOF: For a random variable to be symmetric, it is necessary and sufficient
that its characteristic function be real. By (1.1.6), this is the case if and only if
0 = 0 and (j. = 0. The second statement follows from Property 1.2.2. I
A symmetric stable random variable is strictly stable, but a strictly stable
random variable is not necessarily symmetric. In fact,
Property 1.2.6 Let X ~ Sa(cr, 0, n) with a^\. Then X is strictly stable if and
only ifn = 0.
PROOF: Let X\, X2 be independent copies of X and let A and B be arbitrary
positive constants. By Properties 1.2.1 and 1.2.3,
AX{ + BX2 ~ Sa(a(Aa + Ba)x'a ,0,p(A + B)).
We must set C = (Aa + Bay'a in (1.1.1). By Properties 1.2.2 and 1.2.3,
CX + D~ Sa(a(Aa + Ba)l'a,f3,ti(Aa + Baf'a + D),
and, therefore, we have AX\ + BX2 — CX + D with D = 0 if and only if /x = 0.
The proof is now complete. 1
Corollary 1.2.7 Let X ~ Sa{<J, 0, /x) with a^l. Then X — (lis strictly stable.
Proof: Use Properties 1.2.2 and 1.2.6. I
Thus, any a-stable random variable with q^I can be made strictly stable by
shifting. This is not true when a = 1, as the next property indicates.
Property 1.2.8 X ~ Si {a, j3, /x) is strictly stable if and only if/3 = 0.
Proof: Let X\ and X2 be independent copies of X and let A > 0, B > 0.
Then, by Properties 1.2.3 and 1.2.1,
AX| + BX2~Si((A + B)a, 0, (A + B)fi- -a(3{A\nA + B\nB)j,
whereas
(A + B)X ~ 5! ({A + B)a, /3, (A + B)n - -<r&(A + B) \n{A + B)j.
Therefore D = 0 in (1.1.1) if and only if AX, + BX2 = (A + B)X, i.e., if and
only if
0(AlnA + B\nB) = /3{A + B) \n(A + B)
for any A > 0, B > 0. It is thus necessary and sufficient that 0 — 0. I

1.2
PROPERTIES OF STABLE RANDOM VARIABLES
13
Corollary 1.2.9 lfXu...,Xn are i.i.d. Sa(a, 0, /z), then
Xx+---+Xn= n1/QX, + n{n - nx'a)
if a ^ I, and
9
Xi + ■ ■ ■ + Xn = nXx + -a0n\nn
7T
ifa= 1.
Corollary 1.2.10 (i) No non-strictly I-stable random variable can be made
strictly stable by shifting.
(ii) Every strictly 1 -stable random variable can be made symmetric by shifting.
Of the four parameters a, a, 0 and /z, the parameter /z is the least important
because it affects only location. We shall often assume for simplicity that iz = 0.
Let us now focus on the skewness parameter 0. As noted in Property 1.2.4,
X ~ Sa(a, -0,0) <=$■ -X ~ Sa(a,0,O). The distribution Sa(a,0,O) is said
to be skewed to the right if 0 > 0 and to the left if 0 < 0. It is said to be totally
skewed to the right if 0 = 1 and totally skewed to the left if 0 — — 1.
This terminology is somewhat misleading because it really refers, not to the
support of the distribution, but to the parameters P and Q in (1.1.7) that weigh
the L€vy measure. In the case a < 1 and 0 = 1, however, the support of the
distribution Sa (a, 1,0) is in fact the positive half-line. The following proposition
establishes that the support is contained in the positive half-line by representing the
random variable X ~ Sa{a, 1,0) as a limit of a sequence of positive compound
Poisson random variables. The random variable X ~ Sa(a, 1,0) with 0 < a < 1
is called a stable subordinator.
Proposition 1.2.11 Fix 0 < a < 1, 6 > 0 and let Ng be a Poisson random
variable with mean EN& = 6~a and let Ys,k, k = 1,2,..., be i.i.d. positive
random variables, independent ofNs, with distribution
P(Y > ^ - / 5°X~a ifX>6>
Then the compound Poisson random variable
x6 = j2Yw
k-l
converges in distribution as 8 —» 0 to the stable subordinator
X~Sa{o, 1,0)

14 STABLE RANDOM VARIABLES ON THE REAL LINE 1.2
with
aa =r(l-a)cos(7ra/2).
Moreover, the Laplace transform ofX is given by
£e-7X = e-aQTa,7>0, (1.2.3)
with a > 0 and
aa = T( 1 - a) = aa / cos(7ra/2). (1.2.4)
Proof: For any complex number r,
Er»< = ^exp(-rQ)^-P- = exp{5-°(r - 1)}.
fe!
Therefore
Ecxp{i6X6) = E E[exp(ieX6)\Ns}
= E[Eexv(i6Y6tl)}Ne
= exp{5-a(£;exp(i6iy«il) - 1)}
= exp{5-Q f 6aa\-(a+1){eiex - l)dAJ.
Letting 6 —» 0, we obtain
lim £exp(i0X«) = exp{ f° a\-^+1\eiBX - 0^}. (1.2.5)
Evaluating this integral (see the Appendix of Feller (1966), Chapter XVII.4), we
obtain
lim Eexv(i6X$)
6-+0
= exp|-|6»|Qr(l - a) (^cos — - i sign (0) sin — J J
r 7ra / ,„. 7ra\ "i
= exp|-|6»|ar(l - a) cos — (^1 - t sign (9) tan —J },
which proves that Xs converges in distribution to
r(l-a)cos—J ,l,0j.
Similarly, if 7 > 0,
£exp(-7X5) = exp{rQ /"°° 6aa\-la+l\e-~<x - l)d\}.

1.2 PROPERTIES OF STABLE RANDOM VARIABLES 15
Letting 6 -+ 0,
Eexp(--yX) = «p{/°° aA~<a+,)(e-^ - l)dAJ
/-OO
= exp|7tt / ax-l*+1\e-x - i)dz}
= exp|-7Q / x^e-^dxj
= exp{-7Qr(l-a)},
which establishes (1.2.3). 1
Remarks
1. Relation (1.2.5) is the Livy-Khinchine representation (1.1.7) with Q = 0.
(The term /0°° j^-^s in (1.1.7) converges for a < 1 and can thus be
absorbed in the constant M.)
2. Proposition 1.2.11 shows only that the support of a Sa(a, 1,0) random
variable, 0 < a < 1, is contained in E+. Exercise 1.14 shows that the
support is in fact the whole of E+.
3. Proposition 1.2.11 shows that the Laplace transform Ee~lX, 7 > 0 of
X ~ Sa{a, 1,0), 0 < a < 1, exists and equals e~a°7°'. Because of
the rapid decay of the left tail in the distribution of X when 1 < a < 2
(see (1.2.11) and (1.2.12) below), the "Laplace transform" exists for all
0 < q < 2 (Gupta & Waymire 1990).
Proposition 1.2.12 The "Laplace transform" Ee~~iX, 7 > 0, of the random
variable X ~ Sa(<7,1,0), 0 < a < 2, a > 0, equals
and
Ee"lX = exp< cr-7 ln7 > ifa=l.
PROOF: Use Proposition 1.2.11 if 0 < a < 1 and Exercise 1.15 if 1 < q < 2 to
establish the result. I
Remark. The constant -<ra(cos ^)_1 is negative if 0 < a < 1 and is positive
if 1 < a < 2. It equals a2 when a = 2.
Random variables that are totally skewed to the right can be regarded as basic
building blocks because of the following:

16 STABLE RANDOM VARIABLES ON THE REAL LINE 1.2
Property 1.2.13 Let X have distribution SQ(a,J3,0) with a < 2. Then there
exist two i.i.d. random variables Y\ and Y2 with common distribution Sa(a, 1,0)
such that
d /l + 0\ '/<* f\ - 3\ 'A*
*=( 2 ) y,_( 2 ) Y*?a*l> (L2-6)
and
(1.2.7)
if a = 1.
Proof: This is a direct consequence of Properties 1.2.1, 1.2.2 and 1.2.3. (See
Exercise 1.17.) I
Since —Yi ~ SQ(o, —1,0) is concentrated on the negative real line when
a < 1, we have, by (1.2.6),
Property 1.2.14 For a < 1 and fixed a, the family of distributions Sa(a,P,0)
is stochastically ordered in —1 < (3 < 1; i.e., ifXp ~ Sa{a, fi, 0) and Pi < @2,
then P{Xpl > x} < P{Xp1 > x) for all x. Moreover, SQ((T,/3,0) has support
on the whole real line for —1 < /3 < 1.
The figures in Section 1.6 below show that when a > 1, the family
{Sa(a,P,0), — 1 < /? < 1} is no longer stochastically ordered. Note also
that the support of Sa (a, (3,0) is the whole real line even for /? = ± 1. The tails of
the distribution, however, are affected by the skewness parameter /? as Property
1.2.15 below indicates. Property 1.2.15 concerns the asymptotic behavior of the
tail probabilities P{X > A} and P{X < -A} as A -► oo.
In the Gaussian case a = 2,
P(X < -A) = P(X > A) ~ ^e-^>
as A -> oo (Feller 1957). When a < 2, however, the tail probabilities behave
like A~a. (If oa and 6a are real numbers, we use the notation ax ~ 6a to denote
lim ax/b\ = 1.)
A—>oo
Property 1.2.15 Let X ~ Sa(a, (3, p.) with 0 < a < 2. Then
( limx^00XaP{X>X} = Ca^aa,
{ (1-2-8)
[ limA^oo\aP{X < -A} = CQx-=&oa,

1.2
PROPERTIES OF STABLE RANDOM VARIABLES
17
where
One can use Theorem 1.4.5 below to obtain this result (Exercise 1.27). One can
also apply a central limit theorem type argument, as in Feller (1971), Theorem
XVII.5.1.
In the case a < 1, Property 1.2.15 follows easily from a Tauberian theorem
and (1.2.6). Indeed, let X ~ Sa(a, 1,0). By Proposition 1.2.12, Ee~~lX =
exp{-aa7Q} with aa = aa / cos ?f. Since
/
Jo
1 — Ee—,x
e~lXP{X > \}d\ = (integration by parts)
o 7
1 -exp{-aQ7°}
7
aa7a-'
as 7 —» 0, we obtain, by the Tauberian theorem (Feller 1971, Theorem XIII.5.4),
x-
T(l - a)cos(7ra/2)
aaCa\~a
as A -* oo. This yields the result for a < 1 and P = 1. We use (1.2.6) to obtain
it for a < 1 and all -1 < P < 1. We can then use Proposition 1.3.1 below to
obtain it for all a > 1 and p = 0 (Exercise 1.28).
As a special case, if X ~ SQ(a, 0,0), then as A —* oo,
P(X>A)~aa^A-a. (1.2.10)
Suppose X ~ 5Q(cr,-1,0). Since /? = -1, Property 1.2.15 gives
lim AaP(X > A) = 0, i.e., P(X > A) tends to 0 faster than A_Q as A -» oo.
What is the true rate? When a < 1, X is totally skewed to the left and hence
P(X > A) = 0 for all A > 0. It rums out, that when a > 1, as A —> oo,
P(X > A)
V^TraCa-l) WQ/ r\ Va£7«^ 7
(1.2.11)

18 STABLE RANDOM VARIABLES ON THE REAL LINE 1.2
where
aa — a I cos— (2 - a) \
When a = 1,
P(X > A) ~ -^ exp M^W"1 _ eU/2<x)A-A (1 2 12)
(see Zolotarev (1986), Theorem 2.5.3.). The same formulas apply to the left tail
when f3 = 1 because P{X < -A) = P(-X > A) and X ~ Sa(a, 1,0) implies
-X~Sa{a,-l,0).
The tail behavior (1.2.8) is a widely used property of a-stable distributions.
Since E\X\T = /0°° P{\X\r > A}<2A, we have:
Property 1.2.16 Let X ~ Sa(cr, 0, fj,) with 0< a < 2. Then
E\X\P < oo for any 0 < p < a ,
E\X\P = oo for any p > a .
The fact that a-stable random variables with a < 2 have an infinite second
moment means that many of the techniques valid for the Gaussian case do not
apply. An added complication stems from the fact that even J5|X|Q is infinite.
When a < 1, one also has E\X\ = oo, precluding the use of expectations.
Property 1.2.17 Let X ~ SQ(a,/3,0) with 0 < a < 2 and 0 = 0 in the case
a = 1. Then, for every 0 < p < a, there is a constant ca^{p) such that
{E\X\ni/p = caffr)*. (1.2.13)
The constant ca^{p) equals (£|X0|p)'/p where X0 ~ Sa(l,/3,0).
PROOF: This follows immediately from the fact that X = aXo. I
One can show (Hardin Jr. 1984) that
(ca,p{p))p
= —1—til ed (l + B2 tan2 — ] cos (- arctanf/3tan — )).
PJoXu-p-i sin*udu\P 2) \a V 2))
Note that for fixed a and p and a ^ 1, E\X\P is an even function of /3 and
increases in |/3|. Note also that if X ~ SaS, 0 < a < 2, then the moment E\X\P
determines the scale parameter a of X and, therefore, the whole distribution.

1.2
PROPERTIES OF STABLE RANDOM VARIABLES
19
Property 1.2.18 Let X ~ Sa(a,0,n). Then
lim(a-r)E\X\r = aCaaa, (1.2.14)
rid
lim(a - r)EX<r> = aCaPaa, (1.2.15)
rTa
where Ca is given by (1.2.9) and a<b> :— \a\b sign (a).
PROOF: Clearly, for any r < a, EXL = /0°° P(X > tylr)dt. By Property
1,2.15, for every e > 0, there is an M = M(e) £ (l,oo) such that, for every
u > M,
uaP(X >u)€ (cj-^aa-e, cJ-^-aa+e).
Fixing an e > 0, we have
limsup(a - r)EXT+
OrMa i-oo
' P{X > tXlT)dt + / P(X > tl'r)dt)
/•OO
= limsup(a - r)r I uaP{X > u)ur~a-ldu,
rTa J M"/r
with a similar relation for the limit inferior. Hence
a[cj~^-aa -e)< liminf(a - r)EXl
1 + /3
p(a-r;jCA+soiOa
rTa
Since this is true for every e > 0, we conclude that
< lim sup(a - r)EXr+ < a(ca X-^-aa + e).
lim(a - r)EXl = acJ-^-cr". (1.2.16)
r^a Z
Similarly,
lim(a - r)EXr_ = acJ—^-Ca. (1.2.17)
rTa Z
Our claim, (1.2.14) and (1.2.15), follows now from (1.2.16) and (1.2.17) and the
obvious relations |A*|r = XL + XL and X<T> = XL- XL. I
Property 1.2.19 When 1 < a < 2, the shift parameter /i equals the mean.

20 STABLE RANDOM VARIABLES ON THE REAL LINE 1.3
PROOF: Let X ~ Sa(a, /3, p,), 1 < a < 2. The random variable X has finite
mean (by Property 1.2.16 in the case 1 < a < 2, and because X is Gaussian when
a = 2.) Moreover, X - p, is strictly stable by Corollary 1.2.7. Let X, and X2
be two independent copies of X. By Definition 1.1.1 of stability and (1.1.2), the
relation
A{XX - /x) + B(X2 - p) i (^ + B")'/^* - /x)
holds for any positive yl and B. Taking expectations of both sides gives
A(EX - p) + B(EX - p) = (AQ + Ba)x'a{EX - p),
and thus EX = p. I
For an alternative proof see Exercise 1.19.
1.3 Symmetric testable random variables
We have seen that X is SaS (symmetric a-stable) if and only if X ~ Sa(a, 0,0)
(Property 1.2.5). Using (1.1.6), we observe that if X is SaS, then its characteristic
function takes the particularly simple form
EexpiOX^e-^W". (1.3.1)
Hence a SaS distribution is characterized only by the scale parameter a. A random
variable X is called standard SaS if cr = 1. Note that a standard SaS random
variable with a = 2 is iV(0,2) because, when a = 2, we have a2 = \ Var X.
SaS random variables will play an important role in the sequel. The following
result shows that one can always transform a Sa'S random variable into a SaS
random variable for any 0 < a < a'.
Proposition 1.3.1 Let X ~ Sa>{a, 0,0) with 0 < a' < 2 and let 0 < a < a'.
Let A be an a/a'-stable random variable totally skewed to the right with Laplace
transform
Eexp(--yA) = exp{-7Q/a'}, 7 > 0,
i.e., A ~ Sa/a>({cosj£)a'/a, 1,0), and assume X and A to be independent.
Then
Z = Al/a'X ~ Sa(a, 0,0).
PROOF: For a real 6, we have, by (1.3.1),
Eexpi6Z = Ee*p{i9A1/a'X}
= E{EtxV{i(eAl'a')X}\A)

1.4
SERIES REPRESENTATION
21
= Eexp{-<ra'\e\a'A}
= exp{-(<7a'|0|Q')Q/a'}
= exp{-aa\9\a}. I
In particular, this implies that if X is a zero mean Gaussian random variable
and if A is a positive j -stable random variable independent of X, then
Z = Al'2X ~ SaS.
This shows that every SaS random variable is conditionally Gaussian.3
A result similar to Proposition 1.3.1 holds for skewed X ~ Sa>(cr',(3',0) if
a' ^ 1, namely
f SQ(a,0,O) if a?l,
Z = A{'a X ~ I
y S\ (ct, 0, fi) if a = 1.
The random variable Z is skewed if a ^ 1 and it has non-zero shift if a = 1. The
parameters in its distribution are complicated functions of a' and /?' (Hardin Jr.
1984); however, in the special case a' < 1 and (3' — 1, the parameter /? equals 1
(see Exercise 1.21).
1.4 Series representation
We wish now to show that an a-stable random variable X with 0 < a < 2 can be
represented as a convergent sum of random variables involving arrival times of a
Poisson process.
Let N(t) represent the number of "customer arrivals" in the time interval [0, t].
The process { N(t), t > 0} is a Poisson process with rate A if the interarrival times
Ti+\ — Ti, i > 1, are independent and exponentially distributed with mean 1/A.
In that case, EN(t) = At, i.e., the rate A equals the mean number of arrivals per
unit time.
Poisson processes have many special properties. Here are two of them. If
each arrival is eliminated with probability 0 < p < 1, independently of the others,
then the resulting process is Poisson with rate A(l - p). If {N\(t),t > 0} and
{iY2(i),i > 0} are two independent Poisson processes, with rates Aj and A2,
respectively, then the superposed process {Ni{t) + N2(t),t > 0} is Poisson with
rate A* + A2 (Ross 1985). These properties make intuitive sense. The second one
turns out to be the main ingredient in the proof of the following result.
3If X ~ N(0,<72), then the SaS random variable Z = Al/2X can be viewed informally as
iV(0, ir2i4), i.e., normal with the random variance <p-A.

22 STABLE RANDOM VARIABLES ON THE REAL LINE 1.4
Proposition 1.4.1 Let {r,} denote the arrival times of a Poisson process with rate
1 and let {Ri} be i.i.d. random variables, independent of the sequence {n}. If
the series
oo
Erf1'"* (1.4.1)
converges a.s., then it converges to a strictly a-stable random variable.
PROOF: Assume that Y^Li TiX,aRi converges a.s. to a random variable X and
set X' = ZZM)-l/aKi and X" = ES.ft')-"0-^ «*ere {ri}, {<},
{R'i}, {R"} are independent sequences and are copies of {ti} and {Ri},
respectively. For any A > 0, B > 0, Aa + Ba = 1, we have
oo oo
ax'+bx" = Y,(A~a<yl/aRi+Y,(B~aTiy/aRi
oo
= Erf"-*. (1A2)
To verify this last relation, note firstly that {A~arl} is the sequence of arrival
times of a Poisson process with rate Aa since the interarrival times A~a(Ti+i -
Ti),i > 1, are i.i.d. and satisfy
P{A-a(ri+i - n) >x} = P{ri+l -Ti> Aax} = e-A"x
for x > 0, i.e., they are exponentially distributed with mean A~a. Similarly,
{B~ar"} is the sequence of arrival times of an (independent) Poisson process
with rate Ba. Now let us superpose these two Poisson processes. To do this,
we list on the same time axis the arrival times of both processes and view the
combined arrival times as being generated by a single process. This superposed
process is Poisson with rate Aa + Ba — 1. If t\ , t2i ... denote its arrival times,
then each rt is either A~a r'm or B~a t'„ for some m > 1 or n > 1. Set Ri = R'm
if Ti = A-aT'm and Ri = R'^ if n = B~ar'^. The resulting T^s are obviously
i.i.d., and thus (1.4.2) holds. The random variable X is strictly a-stable because
AX' + BX" = X. I
The preceding proposition suggests that a stable random variable X can be
represented as infinite series of the type (1.4.1). It is necessary, however, to add
some assumptions on a and on the distribution of Ri in order to ensure that the
series converges a.s.. We will require 0 < a < 2 and that the distribution of
Ri have finite absolute ath moment and be symmetric, so that the resulting X is
SaS. In fact, we will set Ri = ^Wi where Wi — \Ri[ and where e* = ±1 is an
independent random variable denoting the sign of Ri. Also, for convenience, we

1.4
SERIES REPRESENTATION
23
write Ti instead of tj. This is because the ith arrival in a Poisson process is the
sum of i independent exponentials and hence has a gamma distribution. These
comments together with Proposition 1.4.1 motivate the following theorem. We
begin by stating our assumptions.
Let {ei, e2,...}, {W\, Wj,...}, {r~i, r2,...} be three independent sequences
of random variables. Here ei, e2l... is an i.i.d. sequence of Rademacher variables,
i.e.,
_J 1 with probability 1/2,
€i ~ 1 -1 with probability 1/2 .
W\,W2,... is an i.i.d. sequence of random variables (not necessarily positive)
with finite absolute ath moment. Finally, let Ti, r2,... be a sequence of arrival
times of a Poisson process with unit arrival rate, i.e., T{ — Yl)=\ ej> where the
ejS are i.i.d. exponential random variables with Ee-j = 1. The random variables
ri,r2,... are, therefore, dependent and they are not identically distributed. The
random variable Ti has a gamma distribution with parameter i and mean BTi = i.
Theorem 1.4.2 Suppose 0 < a < 2. Then the sum £~, eiT7l/aWi
converges almost surely to a random variable X whose distribution is
Sa({C-lE\Wi \a)1/a, 0,0) where Ca is the constant defined in (1.2.9).
PROOF: We proceed in three steps.
Step 1. Let U\, C/2,... be an i.i.d. sequence of uniform random variables on
(0,1), independent of the sequences {e\, e2,...} and {Wi, W2,...}. Set
Yi = eiU~l/aWi,i= 1,2,.-..
Because of the CiS, Y\, Y2,... is a sequence of i.i.d. symmetric random variables.
Let us compute their tail probability. Letting F\W\ denote the distribution of \W\,
we have for A > 0,
P(\Yi\>\) = P(U-l/Q\Wi\>\)
P{Ui < A-a|Wi|a)
P{Ui < X~awa)Fm(dw)
Jo
l-X
f
Jo
/•A /-oo
= / \~awaFm(dw)+ Fm{dw)
= \~a f waF]w](dw) + P{\Wi\ > A}.
Jo
Therefore
lim XaP(\Yi\ > A) = E\Wi\a + Urn \aP{W\ > A} = E\Wi\a,
X—KX A—OO

24 STABLE RANDOM VARIABLES ON THE REAL LINE 1.4
so that Yi is in the domain of (normal) attraction of a symmetric a-stable random
variable, i.e.,
, n
d
nl/a^Yi
X, (1.4.3)
=i
where X ~ Sa{a,0,0) with a = {C-iE\Wi\a)l/c' (see, for example, Feller
(1971), page 581).
Step 2. By writing n"1/" £"=1 eiU~l/aWi in a different way, we shall show
that it has the same limiting distribution as Y,iL\ tiT~x/aWi, where the r\s are
the gamma random variables appearing in the statement of the theorem.
To see the connection between the C/jS and the r\s consider the "arrival"
times in a Poisson process with unit rate. Given Tn+i, the time of the (n + l)th
arrival, the previous n arrivals are uniformly distributed in the interval (0, rn+1).
Since the r\s are ordered, H < T2 < ■ ■ ■ < Fn, the (conditional) distribution
of (ri,r2,... ,rn) is that of the order statistic of n i.i.d. U(0,Tn+l) random
variables. Thus, given rn+i, the vector (lVrn+i, i — 1,2,..., n) is distributed
as Gn, the order statistic distribution of n i.i.d. U(0,1) random variables.
We now proceed by taking advantage of two helpful features. Firstly, the
conditional distribution of (rVrn+i, i = 1,2,..., n) given Tn+i is equal to the
unconditional distribution. (This is because the distribution Gn does not depend
on rn+i.) Secondly, any permutation of the UiS in 5Z"=i eiU~l'aWi preserves
the distribution since the 6iWts are i.i.d.. This means that we can replace the
UiS in that sum by the ordered UiS without affecting the distribution of the sum.
Consequently,
Using (1.4.3), we obtain
f^T") Q^£ir7,/o,Wi4x~5a(<T,0,0). (1.4.4)
Step 3. It remains to show that the left-hand side of (1.4.4) converges a.s. to
ESi^rrl/awi.
By the strong law of large numbers, the event A = {lim^oo ±Tn = 1} D
{T\ > 0} has probability 1, since rn is a sum of n i.i.d. exponential random
variables with mean 1. To prove that £°1, eif,- a Wi converges a.s., it is enough
to show that it converges a.s. for each fixed sequence {Tj} belonging to the event
A. Fix such a sequence. Then the summands CiT" 'aWi are independent and
C\i < Ti < Cii for some positive constants C\ and C2 and i = 1,2,.... We can

1.4
SERIES REPRESENTATION
25
apply the three series theorem (Feller 1971, Theorem IX.9.3). For each A > 0,
we have:
(i)
OO CO
Y,P{\eir-i/awi\>\} = 53p{|wi|a>A°ri}
i=l i=l
OO
< ^p{i^r>AQc,i}
< OO
since E|Wi|a < oo,
(ii)
f; jBCirr,/0ivii{|£irr1/awi| < A} = o
t=l
since each summand equals 0,
(iii)
oo 2
Y,e (e,rr1/awii{|£irr,/0wi| < A})
< c;2/ayr2/a / wH{w < xc\/ail/a}F[Wl{dw)
,.00 rXCl/ax"a
<C x~2/adx / " w2Fm(dw)
Jo Jo
= C [ w2Fm{dw) f x-2/adx
Jo Jx-oc-'w*
/»oo
= C / waFm{dw)
Jo
which is finite (C and C" are positive constants). Therefore £™=1 £,1"^ aWi
converges a.s. to YliL\ £^~ W% as n —*• oo and we conclude that
oo
i=i
Let X and Y be two random variables, possibly defined on different probability
spaces. We say that X has the representation Y if X = Y.

26
STABLE RANDOM VARIABLES ON THE REAL LINE
1.4
Corollary 1.4.3 (Series representation). Any SaS, 0 < a < 2, random variable
with distribution Sa{(T, 0,0) has the series representation
'(W)"'^17"^- (1A5)
Remarks
1. We did not use Proposition 1.4.1 in the proof of Theorem 1.4.2. This
proposition, together with step 3 of the preceding proof establish that X is
(symmetric) a-stable but do not provide the value of a.
2. The distribution of ££, tiT~x,aWt depends only on E\WX \a. The
distribution Wj is immaterial as long as £|Wi|Q < oo. For example, the W{S
can be of (7(0,1) or N(0,1).
3. The series representation can, in principle, be used to generate a SaS
random variable. Although the r\s are not independent, they are easily,
generated as successive sums of exponential random variables. This is
not, however, a good way to generate SaS random variables because the
convergence is too slow. The method of Chambers et al. discussed in
Section 1.7 is much better.
The series representation provides some insight into SaS random variables.
The absolute values |eirr1'aWi| of the successive summands decrease
stochastically as i increases because r, a > TJ > • • ■ . The first summand
eiiy1'aWi is stochastically the greatest in absolute value. Its probability tail
(that is, the tail of its distribution function) has the same asymptotic rate of growth
as an a-stable random variable. Indeed,
/•OO
P{\elr;l/aWl\>\) = / P[Tl<waX-a}Fm](dw)
Jo
= / (1 - e-^iV,!^)
Jo
~ £iw,rA-Q
as A -+ oo. The other summands eiT^^Wi, i > 2 provide the necessary
corrections for the whole sum to have an a-stable distribution.
In fact, Corollary 1.4.3 states that X ~ Sa(o,0,0) can be represented as
X = &JQ JXi eiT-i/aWi with E\Wi\a = oa. Both X and the first term of

1.4 SERIES REPRESENTATION 27
the series, Cai\Tx W\, have the same asymptotic behavior, because on one
hand,
P{X > A) ~ ]rCaa^\-a as A - oo (1.4.6)
(Property 1.2.15), and on the other hand,
P{CxJaexT~x,aWx > A) ~ l-Caaa\-a as A ^ oo. (1.4.7)
The probability tail of the remainder of the series, Y^lLi ei^7 Wi, has a smaller
order. In fact, if E\W\ \a+e < co for some e > 0, then
oo
^IX^rV'^l <oo (1.4.8)
i=2
and, hence,
CO
P(J2eiT-l/aWi>x)=o(X-a)
i=2
as A —> co. But if we only assume E\W\ \a < oo, then (1.4.8) may not hold (see
Exercise 1.25). Nevertheless, one always has
Property 1.4.4
CO
pQTeilV'^Wi > A) = o(A~Q) as A -» oo. (1.4.9)
«=2
PROOF: For any M > 0, write
f^tiT-^Wi = Y^eir^WiUm < M) + J2eir-l/aWi\(\Wi\ > M).
-x'aWi = 53c*rrl/awr<1(|Wi| < M) + ^eir-1/Qi
x=2 i~2 i=2
Since Wt 1(|W*| < M) has moments of all order,
CO
p(5>l71/aWil(|Wi| < M) > A) = o(A~Q)
»=2
follows from (1.4.8). Hence,
oo
IS>_00A°pQ>>rrl/aw; > a)
i=2
co
<IElA-.„Aap(5>I7,/aWil(|Wi| > M) > A)
i=2
oo
< 2 lim AQp(^eirr'/QVVil(|Wi| > M) > AJ
i=l

28 STABLE RANDOM VARIABLES ON THE REAL LINE 1.4
using the relation
P(U + V > A) > P{U >X,V>0) = P{U> X)P(V > 0),
valid for any two independent random variables U and V. Here V = t\V\ and
P{eiWi > 0) > i'. We conclude by (1.4.6) that
oo
I55A^09AapQr>r;-,'aw'i > a) < £|w,|ai(|Wi| > m).
We establish the claim by letting M —> oo. 1
The series representation given in (1.4.5) holds for a SaS random variable.
We now provide a corresponding representation for a skewed a-stable random
variable.
Consider again two independent sequences {W\, Wj,...} and {T\, T2,...}.
Here the sequence W\, W2,... is an i.i.d. sequence of random variables with
E\Wl\a<oo if0<a<2, a^l,
E|Wiln|Wi||<oo ifa=l.
The sequence Y\, Ti,... is denned as above.
Theorem 1.4.5 The series
00
£(rr,/aw,-fcM)
(1.4.10)
t=i
with
if 0<a< 1,
_s_ (j2^ - (j - \)^ EWX i/ a > 1,
i = 1,2,..., converges a.s. to a SQ(cr, /3,0) random variable with
where Ca w f/ie constant defined in (1.2.9) and
£|Wi|asignJVt
0 =
E\Wi\a

1.4
SERIES REPRESENTATION
29
Moreover, in the case a = I, the series
f^Wi-JEW, r^^dx) (1.4.11)
i=l V ,/l/i * /
converges a.s. to a Si (a, P, /x) random variable, where a and P are as above and
li=-EWibi\Wi\.
The proof is essentially similar to that of Theorem 1.4.2 but is much more
technical. It will be given in the following section.
Remarks
1. This theorem can be viewed as a representation theorem. A random variable
X ~ 5Q((T, P, /i), 0 < a < 2 can be represented as
CO
* = M + £(rrI"V«-*ia))
i=l
where the sequence of i.i.d. random variables Wi satisfies
E\W{\a = Cacra
and
E\W1\a%ignWl=CaPcra.
There is, therefore, a wide latitude in the choice of the distribution of the
WjS. This can be useful in applications.
2. To recover the SaS case, replace each W, by ejWj, where P(ej = 1)
= P(ei = —1) = 1/2 and the sequence {£{} is independent of the sequence
{Wi}.
3. The theorem can be used to derive (1.2.8) because the tail of the first term
of the series dominates the tail of the other terms (Exercise 1.27).
4. When 0 < a < 1, the sum Y^Li T~[xlcxWi converges a.s.. For example,
let Wi = 1. Then £~, r~1/Q converges because £?!, i~1/a < oo. By
Theorem 1.4.5, we have
CO
£rrl/a~sa(c-,/M,o)
when 0 < a < 1. Hence, a stable subordinator X ~ Sa(a, 1,0) has the
representation aCa Y1Z] *T •

30
STABLE RANDOM VARIABLES ON THE REAL LINE
1.5
5. The sum £°1, T~ 'a W, generally diverges when a > 1 and the Wj are not
symmetric. The compensation fct- when a > 1 is asymptotically equal, as
i —► oo, to ET~ Wi, since
ET~x,aWi = ET-x/aEWx ~ rllaEWx ~ ^q).
When a = 1, an asymptotic compensation by the mean ET^lWi is also
adequate to ensure convergence of the sum (1.4.11) because, as i —► oo,
ET~xWi ~ r'EW, ~ / ^dz EW*.
6. Suppose a = 1. Although the series (1.4.10) converges to a stable random
variable with zero shift parameter, it has the disadvantage of not being linear
in the WiS. The series (1.4.11), on the other hand, is linear in the Wis and
it will be used later, in the context of stable integrals.
1.5 Series representation of skewed a-stable random
variables
Here, we prove Theorem 1.4.5.
For 0 < a < 2 and i > 1 let
'0 if 0 < a < 1,
da) = \ J,1//*-0 z-2 sinx dx if a = 1,
_ _s_(i^_(i_l)^) if l<a<2.
We observe firstly that
P(£(r71/Q - C^a)) converges) = 1. (1.5.1)
Indeed, (1.5.1) is trivial when 0 < a < 1, and, in the case 1 < a < 2, it follows
simply from
£(rr«/» _ C(Q)) = £(rr«/» _ r>/«) + f(r'/a - c|Q))
i=i i=i <=i
and the following lemma.

1.5
SERIES REPRESENTATION IN THE SKEWED CASE
31
Lemma 1.5.1 Let T\,Y2,--be arrival times of a Poisson process with unit arrival
rate. Then for any a > 0,
lp-1/a _ j-l/al
lim sup —! ■—, = a a.s.
i^a/ i-'/«-i/V21nlni
PROOF: Recall that ETi = i and Var T, = i. By the strong law of large numbers,
Ti/i —► 1 a.s. and
lim — jp—-. = hm *-?=—r. = 1 a.s.
i—oo a.-l irj-'l i^co Qj-lJIirH
But by the law of the iterated logarithm,
ir- — i\
lim sup ' ' = 1 a.s.
i->oo v il In In i
The statement of the lemma follows. 1
We will now prove that the series (1.4.10) converges a.s. when a ^ 1, and
that the series (1.4.11) converges a.s. when a = 1.
(a) Case 0 < a < 1. It is enough to prove that YaLi \Wi\i~x/a < °° a.s., and
we do this by using the three series theorem. For any A > 0,
oo oo
J2 p{\Wi\r i/q > a) = J2 p(\w* r > ^Q) < co
i=l i=l
because .E|Wi|Q < oo. Further, for every i > 1,
jE|Wi|i-,/0,i(|wi|i-1/0 < l) < r1/Q / p(|Wi| > x)dx,
Jo
and
OO -i'/a
/•OO
<C P(|W,|Q>y)dy<oo
because 1?| Wi |Q < co. Therefore
OO
Y,E{Wi\i-l,al{\W&-l/a < 1)) < °°>
i=i

32 STABLE RANDOM VARIABLES ON THE REAL LINE 1.5
which concludes the proof for the case 0 < a < 1.
(b) Case 1 < a < 2. From (1.5.1), it is enough to prove that the series
£,~i TTx/a{Wi - EW{) converges a.s.. Let Yt = Wt - EWU t = 1,2,...,
and observe that Yu Y2,... are i.i.d., with E\Y\ \a < oo and EYX = 0. Let
* = {^r"=1}n<r'>0}- (L5-2)
It is enough to prove that the series YliL\ r~1/Qyi converges a.s. for any fixed
H, r2,... in A because P(AC) = 0. We use once again the three series theorem.
Let
Ti = r71'aYil(Trx'a\Yi\<l),i=lt2,....
Since the convergence of the sum Y^Zi -^(^7 l*il > A) for any A > 0 follows
as before from E\Y\ \a < oo, it is sufficient to show
oo
Y^ETi <oo (1.5.3)
i-l
and
oo
]TVarTi<oo. (1.5.4)
i=l
Observe firstly, that for each i = 1,2,...,
rr\/a
ETt = r-1/Q / yFyMv)
J-r'/"
-r'/o r°°
/ — l ■ roo
yFYl(dy)-T-l/a yFY[
-oo Jr.'
(dy).
because EY\ = 0. Therefore (1.5.3) follows from the convergence of the series
ES. rr'/a /r~- yFvMy) and Ei=, r"1/a /:£" 2/*V, (<*</). The first series
converges because Tu T2,... belongs to the set A and, for some constant c> 0,
oo -oo /-oo °°
J/.00
f y
0
< C-' / t/aFy,(dj/)<CO.
/o
The second series converges for a similar reason. This proves (1.5.3). For (1.5.4),
it is enough to show J21L\ ETi < °°- For some c < °°'
rl/o
'%(dy)
oo oo rr'.,a
t=l t=l J-ri

1.5
SERIES REPRESENTATION IN THE SKEWED CASE
33
■°° i=l
/oo
y2FYl(dy)cmm{l,y-2+a)<0o,
•oo
because £|Yi|a < oo. This establishes (1.5.4).
(c) Case a = 1. In the notation of the case 1 < a < 2, it is enough to show that
the series 2Si ^T^i converges a.s. for any Fx, Tz, ■ ■ ■ in the set A, and thus we
must again verify (1.5.3) and (1.5.4). Sincetheprecedingproof of (1.5.4) applies
to the case a = 1, only (1.5.3) needs to be proved. Proceeding as before, it is
enough to show
OO -oo
5Zrr' / ypy> (<*i/)< oo.
Since 1*1, r2,... are in A, for some c> 0, one has
OO -oo .OO OO
^r-1 / yFYl(dy) = / yFYl(dy)^rTl^Ti<y)
/■OO
< c-1 / yFyi(dy)(|logy| + l)<oo,
Jo
because E\YX log |Y||| < oo.
We have now shown that the series (1.4.10) converges a.s. when a ^ 1, and
the series (1.4.11) converges a.s. when q = 1. Since
U\ Jit* x V.IA x J
lim EWX / ^pdz = -EW! log |W,|, (1.5.5)
l-°°- Jl/n x
■\W,\/n
EWX I
n-0°- Jl/n
the series (1.4.10) converges a.s. in the case a = 1 as well
It remains to determine the distribution of the series (1.4.10) and (1.4.11). The
argument is similar to the one used in the proof of Theorem 1.4.2. Let U\, f/2, • • ■
be a sequence of i.i.d. random variables, uniform on (0,1), and independent of the
sequence Wi,W2,.... Setting Y{ = U{ l/aWit i = 1,2,..., we immediately
obtain, as in the proof of Theorem 1.4.2,
lim \aP{Yi > A) = E(Wi)%
X—'00

34
RANDOM VARIABLES ON THE REAL LINE
1.5
and
AlunoA«P(y1 < _A) = EiW^Z.
Therefore Y, is in the domain of normal attraction of an a-stable law. In the case
0 < a < 1, we have
where X ~ Sa (a, /3,0), with a and Q as in Theorem 1.4.2. In the case 1 < a < 2,
we have
~ £>, - EYt) 4 X,
where X ~ Sa{a, /3,0), with cr and 0 as above. Finally, in the case a = 1,
- Y" Yi-n£ sin ^-4> X,
where X ~ Si (a, /3,0), with «7 and (3 as above (Feller 1971, Theorem XVII.5.3).
Suppose first that 0 < a < 1. As in the proof of Theorem 1.4.2,
n s -^ \ \/a n oo
n-voTjTYi± (^±i) ^rr,/awi -52rr,/aw4 a.s.
i=l \ n / i=1 i=1
as n —» oo, which shows that the series (1.4.10) has the claimed distribution in
the case 0 < a < 1.
Similarly, if 1 < a < 2,
n-r/aJ2{Yi - EYX) = n-""^ - ^n1"1/^^ Jz
{=1 i=i "*"
i (l^±\/ayr-)/aWi--^—nl~l/aEWi
\ n J /—fl a-\
= (^)'/a J2 (rt"1/a^ - EW^~~X (i{a~l)/a - (< - i)<°-wa))
+ « n(a-D/a ((n-.rn+1)1/Q - l) BWl
a — 1 \ y
oo
- £(I7l/aWi-fc{a))a.s.
i=l
as n ~+ oo, which proves that the series (1.4.10) has the claimed distribution in
the case 1 < a < 2 as well.

1.6 GRAPHS AND TABLES OF a-STABLE DENSITIES AND C.D.F.'S 35
Finally, we consider the case a = 1:
lfv rp . Y\ d r„+! ^ . y00 sinWix
-> Y—nF sin — = —— > T^Wi-El =—dx
nJT[ n n ~t Ji/n x%
T-> n rOQ
" fcrv v.i/< »! /
\ n J J\wt\/n V2
as n —> oo, proving that the series (1.4.10) has also the claimed distribution in the
case a = 1.
This, together with (1.5.5), completes the proof of Theorem 1.4.5.
1.6 Graphs and tables of a-stable densities and c.d.f.'s
Let X be a SQ(1, /3,0) random variable and let
F(x; a,/3) = P(X < x), -oo < x < oo,
and
p(x;a,/3) = ^F(x;a,/3)
denote, respectively, the cumulative distribution function and the probability
density function of X. The figures on the following pages display the cumulative
distribution function F(x; a, /?) and the density function p(x; a, /3).
Tables
Tables for the upper-tail probabilities 1 - F(x;a,/3) and the fractiles x/ =
F-'(/; a, /?) are given in DuMouchel (1971) for 4
a = 0.1(0.1)1.9,1.95,
P = -1(0.25)1.
*a(b)c denotes a sequence from a to c with increments equal to 6.

36
STABLE RANDOM VARIABLES ON THE REAL LINE
1.6
OE'O 020 Ol'O 00
SO 20 I'O 0"0
90 fO Z'O 00

Stable CDFs for varying Betas, with Alpha = 1.5
Beta= 0„
Beta=.25
Beta= .5
Beta=.75
Beta= 1
2 o 2
Stable CDFs for varying Betas, with Alpha = 1.0
Beta= 0„
Beta=.25
Beta= .5
Beta=.75
Beta= 1
Beta=0
Beta=.25
Beta=;5
Beta=.75
Beta= 1
QtphiA r,DFs for varvinq Betas, with Alpha - 0.5
-4
X
0
a
r
ra
en
O
-a
B
</>
00
r
m
ui
ra
>
Z
o
o
b
-J

38
STABLE RANDOM VARIABLES ON THE REAL LINE
1.6
«7 «*«"
CO co to cd cfl
S-.9-.9-.S-.S-
<««
I
90
9'0
vo eo
A}ISU8Q
20
VO
00

1.6 GRAPHS AND TABLES OF a-STABLE DENSITIES AND C.DJF.'S 39
Tables for p(x; a, j3) can be found in Holt and Crow (1973) for
a = 0.25 (0.25) 1.75,
/? = -1 (0.25) 1.
Fractile tables can also be found in Brothers, DuMouchel and Paulson(1983) for5:
a = 0.1 (0.1) 1.9,
P = 0(0.1)1,
/ = 0.001,0.002,0.005(0.005)
0.030(0.010)0.100 (0.025)
0.900(0.010)0.970 (0.005)
0.995,0.998,0.999.
That above-cited article includes also the following extreme upper fractiles:
a = 0.5 (0.1) 1.9,
/? = -1(0.25)1,
/ = 0.9990 (0.0001) 0.9999.
We have reproduced the fractiles in Appendix A for the symmetric case (3 = 0,
with the kind permission of Albert Paulson.
The evaluation of these fractiles is based, for the most part, on Zolotarev's
integral representation for the a-stable cumulative distribution function, which
takes the following form in the SaS, a ^ 1, case:
Proposition 1.6.1 For 0 < a < 2, a ^ 1, 0 < 7 < 7r/2, define
U 1 \ - fsina7\°/('~Q) cos((l - 0)7)
\ C0S7 J C0S7
andletX ~ Sa(l,0,0). Then,forx> 0,
, r/i f P(0 < X < x) if 0 < a < 1,
- exV{-xa/{a-l)Ua(l)}di = {
* Jo I P{X >x) if 1< a < 2.
(1.6.1)
3The definition of j3 in Holt and Crow (1973) and in Brothers, DuMouchel and Paulson (1983)
differs from ours. The j3 indicated here corresponds to our definition.

40
STABLE RANDOM VARIABLES ON THE REAL LINE
1.6
a
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
cQ
0.7978840
0.7669871
0.7362977
0.7048938
0.6719293
0.6366203
0.5981977
0.5559184
0.5090762
0.4569742
0.3989431
0.3343501
0.2626145
0.1832279
0.0957816
ln(CQ/2)
-0.9189392
-0.9584325
-0.9992680
-1.0428553
-1.0907493
-1.1447290
-1.2069812
-1.2802809
-1.3683048
-1.4762756
-1.6120837
-1.7887137
-2.0302154
-2.3901720
-3.0388319
Intercept
-0.9197074
-0.9604577
-0.9851851
-1.0320764
-1.0822671
-1.1372527
-1.1996404
-1.2708688
-1.3556695
-1.4576651
-1.5863361
-1.7489705
-1.9661427
-2.2768235
-2.7849723
Slope
-0.5000001
-0.5998732
-0.7016126
-0.8014147
-0.9012536
-1.0012093
-1.1012519
-1.2016571
-1.3023241
-1.4036502
-1.5053023
-1.6088360
-1.7155793
-1.8305965
-1.9798950
Table 1.1: Ca is defined in (1.2.9). Intercept and slope are for the least squares
regression of ln( 1 — /) versus In Xf where / and Xf are the numbers given in the table for
extreme fractiles in Appendix A.
Proof: This follows from Zolotarev (1986), Remark 1, page 78. I
Zolotarev (1986), in equation (2.2.30) on page 79, gives the following
expression for the a-stable cumulative distribution function at x = 0:
1(1-/32) ;/0<a<l,
I(l+/32^) ifKa<2,
where (h is the skewness parameter appearing in the representation (1.1.12) of the
a-stable characteristic function. Recall that /32 is a monotone increasing function
of 3. Hence, as 8 increases, F(0;, a, B), viewed as a function of 8, decreases for
0 < a < 1 but increases for 1 < a < 2. This is reflected in the figures of the
a-stable cumulative distribution function.
Suppose X ~ SQ(1,0,0). One quick way to estimate a is to evaluate the
slope in the log-log plot of P{X > x) versus x. This technique is based on the
asymptotic relation P{X > x} ~ (Ca/2)x~a (see (1.2.10)). Let us see how
good this technique is, in the best of circumstances, namely when the distribution
of X is known, the probabilities P{X > x] are taken from a table and the xs are
very large. We shall use the entries of the extreme upper fractiles in Appendix
A to plot ln(l - /) versus lnx/. The results are given in Table 1.1, where
F(0;a,p) =

1.7
SIMULATION
41
Figure 1.2: Display in the case a = 0.5 of the least squares regression of ln(l — /)
versus In x/.where / and x/ are the numbers given in the table for extreme fractiles in
Appendix A:
"intercept" estimates ln(CQ/2) and "slope" estimates -a. Even though we are
using extremely large values of x, the technique does not give a good estimate at
a = 1.9. It improves as a decreases and is excellent at a = 0.5, at these large
values of x (see Figure 1.2). Note though, that the smaller the value of a, the
larger the xs involved.
1.7 Simulation
To generate the Cauchy distribution S\(cr,0,n) in (1.1.13), one can generate
crtan7 + ^, (1.7.1)
where 7 is uniform on (-tt/2,-k/2). To generate the Levy distribution
Si/2(cr, 1,/z) in (1.1.15) one can generate
aZ~2 + p.,
(1.7.2)
where Z is N(0,1). (Exercises 1.7 and 1.8.) It is easy also to generate the
distributions 5Q( 1,0,0) and SQ(l, 1,0) for a = 2~k, k > 1 (see Brown and
Tukey(1946)).

42 STABLE RANDOM VARIABLES ON THE REAL LINE 1.7
In general, however, the generation of stable random variables is quite an
involved affair. For a good discussion see DuMouchel (1971) and Chambers,
Mallows, and Stuck (1976). One common procedure is based on the following
result:
Proposition 1.7.1 Let 7 be uniform on (—it/2, ir/2) and let W be exponential
with mean 1. Assume 7 and W independent. Then
sincry /cos((l - a)-y) V1 a''a
X~ (cos-yy/* { W ) (L73)
«Sa(l,0,0).
Proof: When 7 takes a value in (0,7r/2), the right-hand side of (1.7.3) can be
expressed as
'a(7)\ <•-«»>/«*
where
W
a/(l-a)
(\ a/u—a)
sin err \ ' .. . ,
-^rj cos((l-a)7).
(&)CaseQ<a < 1.
P(0 < X < x) = P(0 < X < x, 7 > 0)
= P(0 < (a(7)/W)(1-a)/Q < x, 7 > 0)
= P{W > a;-Q/(1-a)a(7), 7 > 0)
= £?exp{-x-Q/(I-a>0W}l{7>o}
= - r exp{-a;-Q/(1-a)a(7)}d7-
Now use (1.6.1) to conclude that X ~ SQ( 1,0,0).
(b) Case 1 < a < 2. Start with P(X > x) = P(X > x, 7 > 0) and proceed as
above, making use of the inequality 1 — a < 0.
(c) Case a = 1. Relation (1.7.3) reduces to X = tan7 whose distribution is
Cauchy (Exercise 1.7). I
There is a corresponding result for skewed a-stable random variables (see
Chambers, Mallows and Stuck(1976)). In the Gaussian case a = 2, (1.7.3)
reduces to W1/2 sin 27/cos 7 = 2iy1/,2sin7, which is the Box-Muller method
for generating a iV(0,2) random variable (Box & Muller 1958).
In order to generate Sa(cr,P,^) random variables, it is only necessary, in
principle, to generate 5Q (1,1,0) random variables because, firstly, if X ~ Sa (1, /3,0)

1.7
SIMULATION
43
can be obtained by generating two independent Sa( 1,1,0) random variables
(Property 1.2.13). Secondly, by Properties 1.2.2 and 1.2.3,
aX + /i~ SQ(er,/3,/z) if a ^ 1,
and
2
aX + -Per Incr + y, ~ SQ(cr,P,fj) ifa=l.
7T
Chambers, Mallows and Stuck (1976) describe a method for generating
testable random variables for any 0 < a < 2 and -1 < /? < 1 based on formulas
of the type (1.7.3). To obtain a numerically accurate representation near a = 1,
they use the form (1.1.10) of the characteristic function. They simulate a random
variable Y with characteristic function
EexpiBY = exp{-|0|a + i0wi(0, a,/3)}
where uj\ is defined in (1.1.11). Observe that
f Y ~ Sa( 1,/?,-/?tan *») if a?tl,
4 (1.7.4)
[ y~Sa(l,/3,0) if a=l.
Thus, X = Y + Jl ~ 5Q(1, /?, 0), where
[ /3tan^ if a/ 1,
[ 0 if a = 1.
Observe also that the shift — /J in y can be very large when /3 ^ 0 and a is close
to 1. For example, whereas the support of X is [0, oo) if (3 = 1 and a < 1, the
support of y is [—JJ, oo) and may start far to the left of 0 if a is close to 1. The
advantage of Y over X is that the distributions of Y vary continuously as a and
/3 vary.
The Fortran program rstab (for "random stable") which generates Y is listed in
Chambers, Mallows and Stuck (1976). S-PLUS6 is a powerful statistical package
which contains an updated version of rstab.
Ten thousand realizations of the random variable Y denned in (1.7.4) were
simulated using the rstab program in S-PLUS, in the cases a = 1, 0 = 1 and
q = 0.99, P = 1. The following figures show the scatter plots, the empirical
cumulative distribution function and the corresponding histograms (empirical
density functions). Although most realizations of Y are small, a number of them
6S-PLUS is marketed by StatSci, a division of MathSoft, Inc. It is based on S which is marketed
by AT&T.

44 STABLE RANDOM VARIABLES ON THE REAL LINE 1.7
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46 STABLE RANDOM VARIABLES ON THE REAL LINE 1.7
are extremely large. Untruncated histograms provide no detail because of their
scale. It is necessary to truncate the extremely long tail of the histogram in order
to obtain a clearer picture of its shape at small values. Notice the continuity of
the distribution of Y around a — 1. The probability density function of Y is
supported on (-co, oo) at a = 1 and on [—63.66, oo) at a = 0.99. Nevertheless,
no realization, in either case, was smaller than —3.
We include a program for generating the a-stable random variable Y in (1.7.4).
It is called rstable and is an improved version of the rstab program listed in
Chambers, Mallows and Stuck (1976). It was supplied by John M. Chambers and
adapted by John Nolan. It is reproduced here with their kind permission.
The program rstable listed below requires the following inputs: ALPHA,
BPRIME, U and W. ALPHA is a and BPRIME is B (as in (1.1.10)), U is a
pseudorandom number uniformly distributed on (0,1), and W is a pseudorandom
number with the exponential distribution P[W < x] = 1 — e~x, x > 0. (To
generate W, generate - In U', where U' is uniform on (0,1), independent of U.)
The output is a pseudorandom number distributed as Y in (1.7.4).
C rstable — random stable standardized form
C
C Adapted from a file supplied by Chambers, et al.
C by John Nolan on 5/18/92.
C That file was converted from RATFOR to FORTRAN
C and all variables and functions have been explicitly typed.
C
C Changed to double precision on 3/15/93 by Vadim Teverovsky,
C mostly for ease in dealing with other routines and S-PLUS.
C
double precision function rstable(alpha,bprime,u,w, st)
C arguments ..
C alpha : characteristic exponent
C bprime : skewness in revised parameterization
C u : uniform variate on (0.,1.), for example from a
C. uniform pseudo-random number generator
C w : exponentially distributed variate
C st: return value
double precision alpha,bprime,u,w,phiby2,thrl,a,eps,piby2,b,
* bb,tau,a2,a2p,b2,b2p,alogz,tan2d,z,d2d,d,st
C double precision da,db
data piby2/l.57079633/
data thrl/0.99/
eps = 1.0-alpha
C compute some tangents
phiby2 = piby2*(u-0.5)
a = phiby2*tan2d(phiby2)

1.7 SIMULATION
bb = tan2d(eps*phiby2)
b = eps*phiby2*bb
if (eps.gt.(-0.99))
* tau = bprime/(tan2d(eps*piby2)*piby2)
if (eps.le.(-0.99))
* tau = bprime*piby2*eps*(l.-eps)*tan2d((l.-eps)*piby2)
C compute some necessary subexpressions
C if phi near pi by 2, use double precision.
C if (a.gt.thrl) then
C double precision
C da = dble(a)**2
C db = dble(b)**2
C ' a2 = l.dO-da
C a2p = l.dO+da
C b2 = l.dO-db
C b2p = l.dO+db
C else
C single precision
a2 = a**2
a2p = l.+a2
a2 = l.-a2
b2 = b**2
b2p = l.+b2
b2 = l.-b2
C endif
C compute coefficient
z = a2p*(b2+2.*phiby2*bb*tau)/(w*a2*b2p)
C compute the exponential-type expression
alogz = dlog(z)
d = d2d(eps*alogz/(1.-eps))* (alogz/(1.-eps))
C compute stable
rstable = (1.+eps*d)*2.*((a-b)*(1.+a*b)-phiby2*tau*bb*
* (b*a2-2.*a))/(a2*b2p)+tau*d
st = rstable
return
end
C d2d evaluate (exp(x)-l)/x
double precision function d2d(z)
double precision z
double precision pl,p2,ql,q2,q3,pv,zz
data pi,p2,ql,q2,q3/.84006 68525 36483 239 d3,
* .20001 11415 89964 569 d2,
* .16801 33705 07926 648 d4,
* .1801 33704 07390 023 d3,l.d0/
C the approximation 1801 from hart et al (1968, p. 213)
if (abs(z) .gt.0.1) then
d2d = (exp(z)-l.O) /z
else
zz = z*z
pv » pl+zz*p2
d2d = 2.d0*pv/(ql+zz*(q2+zz*q3)-z*pv)
endif

48
STABLE RANDOM VARIABLES ON THE REAL LINE
1.7
return
end
C mytand tangent function
double precision function mytand(xarg)
logical neg,inv
double precision p0,pl,p2,q0,ql,q2,piby4,piby2,pi,x,xarg,xx
data p0,pl,p2,q0,ql,q2/.129221035e+3,-.887662377e+l,
* .528644456e-l,.164529332e+3,-.45 1320561e+2,1.0/
C the approximation 4283 from hart et al(1968, p. 251)
data piby4/.785398163/,piby2/l.57079633/
data pi/3.14159265/
neg = .false,
inv = .false.
x = xarg
neg « (x.It.0.0)
•x = abs (x)
C perform range reduction if necessary
if (x.gt.piby4) then
x = dmod(x,pi)
if (x.gt.piby2) then
neg = .not. neg
x = pi-x
endif
if (x.gt.piby4) then
inv = .true.
x = piby2-x
endif
endif
x = x/piby4
C convert to range of rational
XX = X*X
mytand = x*(pO+xx*(pl+xx*p2))/(qO+xx*(ql+xx*q2))
if (neg) mytand = -mytand
if (inv) mytand = 1./mytand
return
end
C tan2d compute tan(x)/x
C function defined only for abs(xarg).le.pi by 4
C for other arguments returns tan(x)/x, computed directly
double precision function tan2d(xarg)
double precision mytand,xarg,pO,pi,p2, qO,ql,q2,piby4,x,xx
data P0,pl,p2,q0,ql,q2/.129221035e+3,-.887 662377e+l,
* .528644456e-l,.164529332e+3,-.45 1320561e+2,1.0/
C the approximation 4283 from hart et al(1968, p. 251)
data piby4/.785398163/
x = abs(xarg)
if (x.gt.piby4) then
tan2d = mytand(xarg)/xarg
else
x = x/piby4
C convert to range of rational
XX = X*X

1.8
EXERCISES
49
tan2d = (pO+xx*(pl+xx*p2))/(piby4*(qO+xx*(ql+xx*q2)))
endif
return
end
Remark. Because S-PLUS allows calls to external C or Fortran functions, it can
use the rstable program given here. It is important that the external program uses
double precision (rstable does that).
1.8 Exercises
Exercise 1.1 Show that Definition 1.1.6 implies Definitions 1.1.1 and 1.1.4.
Exercise 1.2 Show that Definition 1.1.4 implies Definition 1.1.1.
Hint: Suppose X is non-degenerate. Then Cn —+ oo, Cn+\/Cn —> 1 as n —> oo.
For arbitrary constants 0 < a^ < a\, there is a sequence of integers {mn} such
that as n —> oo, Cmn/Cn —» 0,2/a\. Express Y^iZ™* Xi normalized in terms of
a sum of Yh=\ Xi normalized and XT^m-" ^i normalized, and let n —♦ 00.
Exercise 1.3 Prove that wi(6, a,P) in (1.1.11) is jointly continuous in a and /3
but is not continuous in 6, the characteristic function argument. Show that it is
not jointly continuous in 6 and a even outside a neighborhood of 6 = 0. (Let
0-► 6>0 ^ 0 and a-► 1.)
Exercise 1.4 Prove the statements about the representation (1.1.12)
Exercise 1.5 Prove that a-stable random variables have infinitely differentiable
densities.
Hint: Keep integrating the characteristic function.
Exercise 1.6 Find the asymptotic behavior of P(X > X) as A —> 00 for X ~
iV(0, v1).
Exercise 1.7 Let 7 be uniformly distributed on the interval (-f, f). Show that
X = a tan 7 + n ~ S\ (a, 1, fi),
where Si(a, 1,/z) is the Cauchy distribution whose density function is given in
Relation (1.1.13). Use this to check the cumulative distribution function (1.1.14)
of the Cauchy.

50
STABLE RANDOM VARIABLES ON THE REAL LINE
1.8
Exercise 1.8 Let Z ~ AT(0,1). Show that
X = aZ~2 + n ~ Sl/2(a, 1,/jl),
where S\/%{cr, l,n) is the L6vy distribution whose density function is given in
Relation (1.1.15).
Exercise 1.9 Let {X(t), t > 0} be a standard Brownian motion with a.s.
continuous paths. Let Ta denote the first time it reaches the level a > 0. Show that Ta
has the L£vy distribution Si/2(a2,1,0) given in (1.1.15). Use this to check the
cumulative distribution function (1.1.16) of the L6vy distribution.
Hint: By the reflection principle, P[Ta <t]= 2P[X(t) > a). Now use Exercise
1.8.
Exercise 1.10 Prove Property 1.2.1 for a = 1.
Exercise 1.11 Prove Property 1.2.3 for a = 1.
Exercise 1.12 If Xi,...,Xnarei.i.d. SQ(<7,/3,/i) andS„ = X\ H hXn,then
the limiting distribution of
n-x'aSn if0<a<l,
n~l(Sn — 27r~'(7/3nlnn) — fj, ifa=l,
n-'/a(5„-n/x) ifl<a<2,
asn —> oo, is Sa(a,f3,0).
Exercise 1.13 The following theorem (Mijnheer (1975)) characterizes the domain
of attraction of stable distributions. Verify it for Xt ~ Sa (1, /3,0), i — 1,..., n.
(A slowly varying function L at infinity is a non-negative function satisfying
limt^oo L{xt)/L{x) = 1 for all t > 0.)
• Theorem 1.8.1 Let X\,..., Xn be i.i.d. with cumulative distribution
function F. There exist an > 0, bn € R, n = 1,2,..., such that the distribution
ofa^[{Xi +■■■ + Xn) - 6n] converges as n -> oo to Sa{\,(3,0) if and
only if both
(i) xa[\ - F(x) + F(-x)] — L(x) is slowly varying at infinity.
W,+r(filA(-,)^¥Ms-^oc,
The an must satisfy
_, , . r(l-a)cos(7ra/2) ifO < a < 1,
lim ^M = \ 2/n ifa=l,
^§5^ t cos ^ | if\<a<2.

1.8
EXERCISES
51
The bn may be chosen as follows:
{0 forQ<a<l,
na" I^L sin(x/an)dF(x) for a - 1,
n X!^ xdF{x) forl<a<2.
In all cases, an = n1/,QI/o(n) w/iere Lq is slowly varying at infinity.
Exercise 1.14 This exercise complements Proposition 1.2.11, which shows that
the support of a totally skewed to the right a-stable random variable, 0 < a < 1
(i.e., a stable subordinator) is contained in R+. Here, the goal is to prove that this
support actually coincides with R+.
The support of a random variable X is defined as the smallest closed set A
such that P(X G A) = 1.
LetX ~ Sa(a, 1,0) withO < a < 1. Let/be the continuous density function
of X. Prove that f(x) > 0 for every x > 0 by going through the following steps:
(i) Let £ = {x G R: f(x) > 0}. Show that for any A,B > 0 such that
Aa + Ba = 1, one has
AZ + SI = Z,
where AZ + BY, = {Ax + By:x&X, ye £}.
(ii) Conclude that either E = R or Z = R+.
(iii) Use Proposition 1.2.11 to conclude that L = R+.
Exercise 1.15 Prove Proposition 1.2.12 forO < a < 2.
Hints:
1. Using the Schwarz reflection principle (Ahlfors (1979), p. 172), prove the
following lemma:
Let f be a complex-valued function defined on D n R ^ 0, where D is
a region and R is the real line. If f is continuable to D+ = {z S D :
1m z > 0}, i.e. if there exists a function f analytic in D+, continuous
in D+ U (D n R), which satisfies f{t) = f(t) for t G £> n R, rten fto
continuation is unique.
2. Show, using (1.2.8), and (1.2.11), (1.2.12) for /3 = 1, that Ee~iX exists
and is continuous for Re £ > 0, and is analytic for Re £ > 0.
3. ForO < q < 2,a ^ l,/3 = 1 andS > 0 the characteristic function (1.1.6)
can be written as $(9) = Eexp(i9X) = exp{-oa6ae~i:ta/cos^)
and can be extended to an analytic single-valued function <f>(z) =
exp(-crQ(—iz)a/ cos zf-), where z belongs to the whole complex plane
with the exception of some half-line (say arg z = — |7r). Verify that using
formula (1.1.6) for 9 < 0 you obtain the same extension.
iVi. t.

52
STABLE RANDOM VARIABLES ON THE REAL LINE
1.8
4. By the above lemma, the functions <f>(z) and £e~(-iz)x coincide for
Im z > 0.
5. Substitute ~iz = 7 in the formula for 4> to establish Proposition 1.2.12 for
6. For a = 1 start with the other part of (1.1.6).
7. The proof is even simpler for a = 2.
Alternative method: Establish Step 2. By Theorem 11.2.1 of Lukacs (1970), p.
311, the Levy-Khinchine representation (1.1.7) of the characteristic function of
X holds for Im z > 0. Show that the passage from the Livy-Khinchine formulas
to the form (1.1.6) is also valid for z ~ it, t > 0. When a = 1, you need also to
prove .
cosx — e~x ,
dx = 0.
x
Exercise 1.16 If X is a-stable then the random variable Y — ex is called log-
stable. Y is log-normal if a = 2. Suppose 0 < a < 2 and show the following. If
/? 7^ — 1, then P(Y > y) is asymptotically proportional to (lny)Q_1 as y —> 00,
and consequently, Y has no finite moments. On the other hand, if f3 = — 1, all
moments of Y are finite.
Log-stable distributions with /3 = —1 are used in modeling multifractals,
partly because of the finiteness of their moments. See for example Evertsz and
Mandelbrot (1991), (1992) and Schertzer and Lovejoy (1993).
Exercise 1.17 Prove Property 1.2.13.
Exercise 1.18 Let X ~ Si (a, /?, 0) with 0 < a < K. Show that there is a finite
constant ^K which depends only on K and a positive number a, independent of
a, /3 and K, such that P(X > 7*-) > a.
Hint: Use Property 1.2.13 and set a = P{YX > 0) • P(Y2 < 0) > 0 where Y and
Yi are i.i.d. Si (1,1,0) random variables.
Exercise 1.19 Using the weak law of large numbers to prove that when 1 < a <
2, the shift parameter (i equals the mean.
Exercise 1.20 Formulas for the density functions of the Cauchy distribution
Si(<7,0,/i) and for the L£vy distribution Si/2{c, 1,/x) are given at the end of
the first section. Use Property 1.2.15 to verify that the normalization constants in
these formulas are correct.
L

1.8
EXERCISES
53
Exercise 1.21 LetX~ $a'(cr', 1,0), and A ~ Sa/a,{aA, 1,0), 0 < a < a' < 1
be independent. Show that Z — Alla X has a Sa{cr, 1,0) distribution for some
<7>0.
Exercise 1.22 Let r\ have a gamma distribution with parameters i and ETiy as
in Theorem 1.4.2.
(a) Find the asymptotic behavior as A —♦ oo of P{TJl,a > A} forO < a < 2.
(b) Let aTTl/aWi be defined as in Theorem 1.4.2. Find the asymptotic
behavior of P{\uT~x,aWi\ > A} as A -> oo.
(c) What are the positive numbers p for which ET^P < oo?
(d) If p > 1, show that there is a constant K such that
ET-p = T{i - p)/{i - 1)! < Ki~p.
Exercise 1.23 Let Y* = e^ ' Wi be defined as in Theorem 1.4.2 and suppose
that q = 2. Find the normalization dn such that d~' ]£?=i ^ converges in
distribution to a non-degenerate random variable Z. What is the distribution of
Z1
Exercise 1.24 Let X\, X2,... be a sequence of i.i.d. random variables and let
p > 0. Show that
(i) E\Xt |p < 00 is and only if lim^^ n~VpXn = 0 a.s„
(ii) S|Xi|p — 00 if and only if limn_^00n_l''p|Xn| = 00 a.s.
More generally, show that for p > 0 and r£R,
(iii) E\Xi |p(log \Xi \)r < 00 if and only if lim,,..*, n-'/p(logn)r/pX„ = 0
a.s.,
(iv)£|X1|p(log|X,|)r = coifandonlyifiImn_00n-1/p(logn)r/p|Xn| = 00
a.s.
Exercise 1.25 How large can the remainder of the series 2Z°12 eJ*7" "^ ^
(i) Suppose that E\Wj\a+t < 00 for some e > 0. Prove that
E\T,7=2*ir7i/awj\a<™-
(ii) Show by counterexample that £J|Wj|0' < 00 does not imply that
£IE^r;l/aw-|«<oo.
Hint: Take WjS such that E\Wj\a < 00 but £|Wj|atog1Wi| = 00 and use
Exercise 1.24.

54
STABLE RANDOM VARIABLES ON THE REAL LINE
1.8
Exercise 1.26 Let {Xi} be independent random variables Xi ~ SQ(cri,/3i,^i),
0 < a < 2. Prove that the series £~ t Xi converges a.s. if and only if £~, ^
converges and YaL\ °f < °°-
Using the three series theorem, prove that the series converges absolutely a.s.
if and only if YX=\ iMi I < oo and
YXL\°i < °° ifO<a<l,
- ESi"i|ln(<7iA0.5)|<oo ifa=l,
. YX=\°i < °° ifa> 1.
Exercise 1.27 Prove (1.2.8) using Theorem 1.4.5.
Hint: Use Exercise 1.22.
Exercise 1.28 Suppose X ~ So(a,0,0). Use Proposition 1.3.1 to show that if
(1.2.8) holds for 0 < a < 1 then it holds for 1 < a < 2.
Exercise 1.29 Let Xa denote a stable subordinator, i.e., Xa ~ Sa(a, 1,0) with
a > 0 and a < 1. Prove that as a —♦ 0,
(Xa)a - |,
where E is an exponential random variable with mean 1.
Hint: Use the series representation for Xa and the fact (prove it) that a sequence
a = {di}?!, in i^.po > 1 satisfies lim ||a|L = Halloo.
' p—*oo

Chapter 2
Multivariate stable
distributions
The multivariate stable distribution, which is the distribution of a stable random
vector, is denned in Section 2.1 by simply extending to M.d the definition of a
stable random variable. Gaussian vectors, however, can also be defined in the
following way: a random vector is Gaussian if and only if any linear combination
of its components is a Gaussian random variable. As in the Gaussian case,
any linear combination of the components of a stable random vector is a stable
random variable, but the converse is not always true. The converse holds when
the linear combinations are either strictly stable or when a > 1. In Section 2.2,
we demonstrate with a counterexample that the converse does not generally hold
when a < 1.
As in the univariate case, multivariate stable cumulative distribution functions
or density functions are not usually known in closed form and therefore one works
instead with the characteristic functions. The characteristic function of ana-stable
random vector is described in Section 2.3. It involves a finite measure T on the
unit sphere of Ed and a shift vector n° which plays a role similar to the shift
parameter in the univariate case. The measure T is called the spectral measure.
It replaces both the scale and skewness parameter that enter in the description of
the univariate stable distribution.
Conditions for strict stability of X are discussed in Section 2.4. When a ^ 1,
X is strictly stable if the shift vector /x° = 0. Conditions for strict stability
when a = 1 are more complicated because they involve the spectral measure
T. Nevertheless, strict stability turns out to be a componentwise property: X is
strictly stable if and only if all its components are strictly stable.
Section 2.4 also contains conditions for X to be symmetric, i.e., for the vectors

56 MULTIVARIATE STABLE DISTRIBUTIONS
X and -X to be identically distributed. In contrast to strict stability, symmetry is
not a componentwise property: X is symmetric if and only if the shift vector fj,°
is zero and the spectral measure T is symmetric (i.e., T gives equal weight to any
two antipodal sets).
We noted in Chapter 1 that every symmetric a-stable random variable has
the same distribution as that of a mean zero Gaussian random variable
multiplied by A1/2, where A is a totally skewed stable random variable. Only some
symmetric a-stable random vectors have a similar property. Those that do are
called sub-Gaussian and they are introduced in Section 2.5. Because they are
obtained by multiplying a mean zero Gaussian vector by A1/2, they inherit the
dependence structure of the underlying Gaussian vector. The multivariate Cauchy
is an example of a sub-Gaussian random vector.
When the spectral measure T is uniform (and jz° = 0) the symmetric
testable random vector has a very special type of sub-Gaussian structure: it is
conditionally i.i.d. Gaussian. Such a structure is used in Section 2.6, in the context
of isotropic complex-valued random variables. Although we are interested in real-
valued random objects, we will later encounter an important class of real-valued
stochastic processes (the so called "harmonizable processes") which are typically
defined in terms of complex random variables. A complex random variable can
be regarded as a two-dimensional random vector whose components are the real
and imaginary parts.
When a = 2, the random vector is Gaussian and its dependence structure
is completely specified by its autocovariance function. There is no such simple
description when a < 2. Covariances do not exist, but when a > 1, one can
introduce the notion of covariation. We do this in Section 2.7. The covariation
shares some of the properties of the covariance. Unfortunately, it is neither
symmetric nor additive in its second component. However, it becomes additive
in its second component when the random variables that are being summed are
independent. This feature will be useful later, in the context of stable stochastic
integrals.
In Section 2.8, we introduce covariation norms. Like the covariance, the
covariation gives rise to a norm. The covariation norm of an a-stable random
variable turns out to be identical to the scale parameter of that random variable.
It metrizes convergence in probability, i.e., the covariation norm of a sequence of
a-stable random variables tends to zero if and only if that sequence converges to
zero in probability.
In Section 2.9, we introduce James orthogonality. Two zero mean Gaussian
random variables have zero covariance if and only if they are independent. If
two symmetric a-stable random variables with 1 < a < 2 are independent, then
they have zero covariation, but the converse is not always true. It is possible to

2.1
STABLE RANDOM VECTORS
57
relate "zero covariation" to "James orthogonality." James orthogonality is well
defined in normed vector spaces that possess no scalar product. We define it
in our context by using covariation norms. Unfortunately, it is not a symmetric
relation; that is, if X is James orthogonal to Y, then Y is not necessarily James
orthogonal to X. Although not as versatile as the usual notion of orthogonality,
James orthogonality is often valuable. It is used, for instance, to characterize the
family of sub-Gaussian random vectors.
The codifference, another measure of bivariate dependence, is introduced in
Section 2.10. Like the covariation, the codifference reduces to the covariance
when a — 2. But whereas the covariation may not be defined for a < 1, the
codifference is defined for all 0 < a < 2. It is, moreover, symmetric in its
arguments and the codifference of a SaS random vector is non-negative definite.
We will encounter more properties of multivariate stable distributions in later
chapters.
2.1 Stable random vectors
The definition of stability in Rd is analogous to that in M1.
Definition 2.1.1 A random vector X = (X\ ,Xi,..., Xd) is said to be a stable
random vector in M.d if for any positive numbers A and B there is a positive
number C and a vector D G Rd such that
AX(" + BX<2> = CX + D, (2.1.1)
where X^ and X^ are independent copies of X.
Instead of saying "X = {X\,..., Xj) is a stable random vector in Rd," we will
often say "X{, X2, ■ ■ ■, Xd are jointly stable", or "X has a stable distribution in
Rd," or "the distribution of X is multivariate stable."
The vector X is called strictly stable if (2.1.1) holds with D= 0 for any A > 0
and B > 0. The vector X is called symmetric stable if it is stable and satisfies in
addition the relation
P{X e A} = P{-X e A}
for any Borel set A of Ed. As in K1, a symmetric stable vector is strictly stable.
The preceding definitions impose conditions on the joint distribution of
(Xi,X2,..-,Xd). What do they imply about the individual components
X\, X2, ■.., Xdi. If X is a stable random vector, are its components stable random
variables? Are linear combinations of its components stable random variables?
The following theorem provides an answer.

58 MULTIVARIATE STABLE DISTRIBUTIONS 2.1
Theorem 2.1.2 Let X = (X\,..., Xd) be a stable (respectively, strictly stable,
symmetric stable) vector in Rd. Then there is a constant a 6 (0,2] such that, in
(2.1.1), C = {Aa+Ba)^a. Moreover, any linear combination of the components
ofX of the type Y = ELt hXk is an a-stable (respectively, strictly a-stable,
symmetric a-stable) random variable.
Proof: Let X(1) and X^2) be independent copies of X, let b = (fy, fe,..., bd)
be a vector in Rd, and define V, = £)Li bkX^ and Y2 - £fc=i bkx£\ The
random variables Yj and Y2 are independent copies of the random variable Y.
Now fix A > 0 and B > 0. Since X is stable, by (2.1.1), there are a C > 0 and
D = (D,,..., Dd) such that AX™ + BX.W = CX + D. When two vectors are
equal in distribution, linear combinations of their respective components are also
equal in distribution, as can be seen by comparing the characteristic functions.
Hence,
d
AY + BY2 = J2b^AXil) + BX™)
fc=i
d
= £>(CXfc + A0
k=l
d d
= Cj2hXk + Y,bkDk .
= CY + (b,D),
where (b, D) denotes the scalar product of the vectors b and D. This shows
that Y is a stable random variable. By Theorem 1.1.2, there is a constant a =
a(b) € (0,2] such that C = {Aa + 5a)1/o. In fact, a is independent of b,
because if a' is the index resulting from a vector b' =£ b, then we would have
{Aa + BaYla = (Aa' + Ba')l/a' for all A > 0 and B > 0, which is impossible
unless a = a'. I
As in the one-dimensional case, we obtain (see Exercise 2.2):
Corollary 2.1.3 A random vector X is stable if and only if for any n>2, there
is ana € (0,2] and a vector Dn such that
X0> + X<2> + • • • + X<"> = n'/QX + Dn, (2.1.2)
where X(1\ X^,..., X^ are independent copies ofX.
These results motivate the following definition:

2.1
STABLE RANDOM VECTORS
59
Definition 2.1.4 A random vector X in Ed is called a-stable if (2.1.1) holds with
C = (Aa+ Ba)xla, or, equivalently, if (2.1.2) holds. The index a is the index of
stability or the characteristic exponent of the vector X.
The second part of Theorem 2.1.2 states that if X is an a-stable vector, then
all linear combinations (b, X) = £)i=i biXi are a-stable random variables.
Is the converse true? If all of linear combinations of the components of a
random vector X are a-stable, is the vector X necessarily stable? The question is
a natural one in view of the fact that in the Gaussian case a = 2, it is well known
that the answer is yes. Unfortunately, when a < 2, the answer is, in general, no
as we will show in the next section. It turns out, however, that the answer is yes
if all the linear combinations are strictly stable, or if a > 1.
Theorem 2.1.5 Let ~K.be a random vector in Rd.
(a) If all linear combinations Y = Ylk=\ ^fc-^fc nave strictly stable
distributions, then X is a strictly stable random vector in Rd.
(b) If all linear combinations are symmetric stable, then X is a symmetric
stable random vector in M.d.
(c) If all linear combinations are stable with index of stability greater or equal
to one, then X is a stable vector in \Wd.
PROOF: The index of stability of a linear combination Yb = ^t=i bfc-Xfc =
(b, X) may a priori depend on b. We first show that if all linear combinations Yb
are stable, then those that are non-degenerate have the same index of stability.
Suppose that this is not so, that there are non-zero vectors b 6 Md and c e M.d,
such that Y"b = (b, X) and Yc = (c, X) are non-degenerate random variables,
and such that Yb has index of stability a\ and Yc has index of stability a2 with
0 < a>\ < ot2 < 2. Let p\, /?2 be any non-zero real numbers and set a = p\h+p2C.
Consider the random variable Ya. By our assumptions, Ya must be stable with
some index of stability, say 03. Then for any A > 0,
P(|Ya|>A) = P(|(a,X)|>A)
= P(|p,(b,X) + ft(c,X)|>A)
= P(\piYb + P2Yc\>\)
> P(\PlYb\ > 2A) - P(\pzYc\ > A)
~ const. A_Q| - const. A-Q2
~ const. A~°"
as A -> 00. (If a2 = 2, replace A~°2 by A-1 exp{-A2/4cr2}.) This implies
a3 < ct\ — min(ai,a2). The index of stability of Ya is, therefore, no larger

60 MULTIVARIATE STABLE DISTRIBUTIONS 2.1
than the smaller of a\ and a2. Now, since Y"b and Yc are non-degenerate, there
is a sequence of non-zero real numbers {An}%Lx, converging to 0 as n —► 00,
such that for all n, Zn — AnY\> + Yc is not equal to zero in distribution. Let
Zn ~ Sa(n)(<r„, 0n, Mn). n = 1,2,.... By the above fact,
<*(") < "i < a2, n = 1,2,
Clearly, Zn -+ Yc in distribution as n -+ 00. Convergence of the real part of the
characteristic function of Zn to that of Yc implies
Km <7*<n>|0|°<B> = a?\6\a\ 8eR,
n—>oo
where a2 is the scale parameter of Yc. Hence limn_K3O0£(n)|0|Q(n)_O!2 =
a%2, #7^ 0. together with a(n) - a2 < ot\ - a2 < 0, implies a2 — 0,
contradicting the fact that Yc is non-degenerate. All non-degenerate linear combinations of
the form (b, X) must therefore have the same index of stability.
To establish (a), assume now that all linear combinations (b,X) are strictly
stable random variables with index of stability a.
Fix any A > 0, B > 0, let Z = AX™ + BX.$\ and W = {Aa + Ba)'/°X,
where X'1' and X^2^ are independent copies of X. We want to show that Z = W.
For any vector b € Rd, we have
(b,Z) = A(b,X^s)) + B(b,X(2)) = (Aa + Ba)^a(h,X) = (b,W).
Therefore, for any b e Rd,
Eexpi(b,Z) = Eexpi(b,W),
showing that the vectors Z and W have identical characteristic functions. Thus,
Z = W, i.e., AXO + BX^ = (Aa + Ba)^aX. This proves that X is a strictly
stable vector in Rd.
(b) Assume that all linear combinations (b, X) are symmetric stable random
variables. Since a symmetric stable random variable is strictly stable, we conclude
by part (a) that X is a strictly stable vector. To prove part (b), one has only to
show that X is a symmetric vector.
Let b be a fixed arbitrary vector in Rd. Since (b, X) is a symmetric random
variable, we have
Eexpi(b,X) = Eexp{-i(b,X)} = £expi(b,-X),
i.e., the characteristic function of X equals that of —X. Hence X = —X,
establishing part (b).

2.1
STABLE RANDOM VECTORS
61
(c) Assume that all linear combinations are stable random variables with index
of stability a > 1.
Suppose firstly a = 1. By assumption, for every b € Rd,
Yb = (b,X) ~ S,(<r(b), /3(b),M(b)) (2.1.3)
for some cr(b) > 0, /3(b) € [-1,1] and /x(b) in E. If crfcl /3fe and fj,k
denote, respectively, a(b), /3(b), ^(b), corresponding to the vector b =
(0,..., 0,1,0,... ,0) with 1 in the kth position, then
Xk ~ Si(ak,Pk,Hk), k= l,...,d.
Let X^[), X^2^,... be i.i.d. copies of X and define, for n > 1,
SW = H EXW) - l(^n){a x /3), (2.1.4)
where a x (3 = {ciPi,crtPz,.. .,ad(3d). Using Properties 1.2.1, 1.2.3 and 1.2.2,
Xk ~ Si{ak,0k,Hk) implies
n
Y,Xk] ~ 5,(rwTfc,ft,nMfc),
-E43' ~ 5,(<7fc,/3fc>/ifc+-(lnn)fffc)9fc)
S<n) 4 Xfc~S,(<rfc,j3fc,/ifc)
for each fc = 1,..., n. Since the distribution of 5j.n does not depend on n, the
sequence {Skn', n > 1} is tight; i.e., for each e > 0, there exists a bounded set
[a, b] such that P(a < Skn) < b) > 1 - e holds for all n. The sequence of vectors
{S(n), n > 1} is therefore tight (Exercise 2.3), and as such, it has a subsequence
{S^ni\ i > 1} converging weakly to a probability measure on Rd (Billingsley
1986, Theorem 29.3). In particular, (b, S(n;)) converges weakly forevery vector
b = (6,,..., 6n) in Rd. But (2.1.3) and (2.1.4) imply
d
(b,S<n>) ~ S,(a(b),/3(b),M(b) + -(lnn)(a(b)/3(b) -£>**&)).
(2.1.5)
In order for (b, S(ni)) to converge weakly, the coefficient of Inn must be zero,
i.e.,
d
<r(b)/3(b) = £ bkVkPk for every b e Rd. (2.1.6)
fc=i

62 MULTIVARIATE STABLE DISTRIBUTIONS 2.1
Now (2.1.3), (2.1.4) and (2.1.6) imply (b,S<B>) = (b, X) for every b 6 Rd and
every n > 1. Therefore X = S(n) for every n > 1. Thus, X satisfies (2.1.2) with
a = 1 and so X is a 1-stable random vector.
A similar proof holds for the case 1 < a < 2. Setting
S(B) = (^Ex°))-»,-,/a'* + M,
we have
(bISW)~Stt(a(b),j8(b),n,-,/°(/x(b)-^6tMfc)+5];6t/ifc).
fc=i fc=i
This time, the result follows from the fact that n1-1/Q -K»asn->oo. I
Remarks
1. The proof of part (c) fails when 0 < a < 1 because nl~l/a —+ Oasn —» oo.
2. Here is an alternate proof of part (c) when 1 < a < 2. The idea is
to make use of part (a). Observe that E\Xk\ < oo for fc = 1,2,..., d.
Indeed, Xk is a particular linear combination of the components of X,
and since, by assumption, all linear combinations are stable with index
greater than one, they all have a finite mean (Property 1.2.16). Let Yb =
Ek=i bkXk, let X = (XUX2, ...,Xd) where Xk = Xk - EXk, k =
1,..., d are zero mean random variables, and let Yb = Yfk=\ bkXk- By
the linearity of the expectation,
d d
Yb = Y,bkXk-EY,bkXk, (2.1.7)
jt=i fe=i
i.e., Yi, = *b ~ EYb- Yt> is stable by assumption and therefore, Yb is strictly
stable by Property 1.2.19 and Corollary 1.2.7. This implies by part (a) that
X is strictly stable and hence X = X + {EXi, EX2, ■■■, EXd) is stable.
3. The key step in the preceding proof of part (c) was the use in (2.1.7) of the
linearity of the expectation in order to reduce the general case to the strictly
stable case. When a = 1, compensation by shifting is not enough to make
the components strictly stable (see Corollary 1.2.10). When a < 1, one can
still make the components strictly stable by shifting (Corollary 1.2.7), but
the shift for each component is not equal to the expectation (the expectation
is infinite when a < 1). As a result, linearity is lost: the shift for a given
linear combination is not equal to the linear combination of the shifts. This
means that a computation similar to (2.1.7) is not possible when a < 1.

2.2
A COUNTEREXAMPLE FOR 0 < a < 1
63
When q < 1, one can construct a counterexample which shows that the
stability of all the linear combinations (b, X) does not ensure the stability of the
vector X.
2.2 A counterexample for 0 < a < 1
We shall exhibit a counterexample due to David J. Marcus which demonstrates
that there exists a non-stable vector X = (X\,X2) in R2 such that all linear
combinations of its components are a-stable random variables with a < 1.
Fix 0 < a < 1 and p > 0 and let
Y(0i, 02) = exp{ -Ta + ipr cos(3<£)} (2.2.1)
where
6\ = r cos <f> and 62 — r sin cf>.
For sufficiently small p > 0, ^(#1, #2) is the characteristic function of a random
vector X = (Xu X2) in R2 (Marcus 1983).
We now proceed in two steps.
Step 1. We first show that for any vector b = (^1,62) in R2> the linear
combination (b, X) = b\X\ + 62X2 is a a-stable.
The characteristic function of the random variable (b, X) is
Eexp{i0(b,X)} = £exp{i(0b,X)} == "¥{Bb\,9bi)
where 6 is real valued. Setting b\ = rb cos $b, h. — rb sin <£b, we can write
6bx = |0|rbcos(<£b + 7rl(0<O)),
#62 = |6>|rbsin(4>b + 7rl(0<O)),
so that by (2.2.1),
£exp{i6>(b,X)} = exp{-(|0|r6)Q + ip|0|rbcos[3((/>b + 7rl(0<O))]}
■= exp{-|0|arg + iprb\6\ sign 0 cos(3<?>b)}
= exp{-rg|0|Q+i0(prbcos30b)}.
By Definition 1.1.6, this is the characteristic function of
Sa{rb, 0, prb cos 30b), the a-stable distribution with scale parameter rb, skew-
ness parameter equal to 0 and shift parameter pr^ cos 3^b-

64 MULTIVARIATE STABLE DISTRIBUTIONS 2.2
Step 2. We now show that the vector X with characteristic function ^(Oi, 92)
is not a stable random vector in R2 when 0 < a < 1.
Suppose, to the contrary, that that X is a stable random vector in K2. It is then
stable with index a because, by Step 1, any linear combination of its components
is a-stable. Applying (2.1.1) with A = B = 1 and C = (Aa + Bafla = 2'/°,
we obtain
X(1) + X^ = 21/aX + D (2.2.2)
for some vector D = (D\,D2)mR2. (The vectors X.^ andX<2' are independent
copies of X.) Relation (2.2.2) implies that the characteristic function of X i + X2
must equal the characteristic function of 2'/aX + D.
Let 9 = (#1,92) and 6\ = r cos <j) and 92 = r sin <j>. The characteristic function
ofXO+X^is
Eexpi(0,X(1)+X(2)) = £expi(0,X(1))£expi(0,X(2))
= [£expi(0,X)]2
= exp{-2rQ +i2prcos(3<£)},
whereas the characteristic function of 21/,QX + D is
£expi(0,21/oX + D)
= exp{i(0,D)}Eexpi(21/a0,X)
= exp{i(0)D)}Y(21/a01,21/<>02)
= exp{-2r0 + i[p{2l/ar) cos(3<£) + 0,D, + 92D2}}.
Equating the two characteristic functions yields
2prcos(3<j>) = 21/Qprcos(3<M + 0,A + 92D2,
for all real 9\ and 62, i.e.,
(2-21/q)/>cos3<£ = £>|COS0 + -D2sin<£
for all <f> € [0, 2tt). Setting <j> = ?r/2 yields D2 = 0. Therefore,
(2-21/a)pcos3<£ = .Dicostf>, (2.2.3)
for all (j> € [0,27r). Since this is clearly impossible, we conclude that the vector
X = (X[, X2) does not have a stable distribution.

2.3 CHARACTERISTIC FUNCTION OF AN a-STABLE RANDOM VECTOR 65
Remarks
1. The function *¥{6\,d2) given in (2.2.1) is a characteristic function even
when a = 1 (provided that the parameter p > 0 is small enough) but, by
Theorem 2.1.5, X is a 1-stable vector in this case.
2. The family of infinitely divisible distributions includes the stable
distributions. When a < 1, the vector X with characteristic function *F(0i, #2) is
not even infinitely divisible. This follows from the following theorem of
Gine" and Hahn (1983).
Theorem 2.2.1 Let X be a random vector in Rd such that all
linear combinations of its components are stable. If X is also infinitely
divisible, then X is stable.
2.3 Characteristic function of an a-stable random
vector
Let X = (X(, X2,. -., Xd) be an a-stable random vector in Ed and let
d
<M0) = <M0i,02, ■ • • ,0d) = £exp{i(0,X)} = £exp{i^0fc;ffe}
fc=i
denote its characteristic function. (3>Q(0i,..., #<*) is also called the joint
characteristic function of the random variables X\, X2, ■ ■ ■, Xd-)
The expression for <I>a(0) given in the following theorem involves an
integration over Sd = {s: |js|| = 1}, the unit sphere in Ed. Observe that Sd is a
(d — l)-dimensional surface. For example, S\ is the two point set {—1,1} and S2
is the unit circle.
Theorem 2.3.1 Let 0 < a < 2. Then X = {XUX2, ...,Xd) is an a-stable
random vector in Rd if and only if there exists a finite measure T on the unit
sphere Sd o/Ed and a vector jx° in Rd such that:
(a) If a ^ I,
<DQ(0) = exp{- J |(0,s)|Q (l - i sign ((0,s)) tan ™)r(ds) + i(6, p0)}.
(2.3.1)
(b)lfa=\.
<M0) = expi- f \{e,s)\(l+i-sign({e,s))\n\(d,s)\y(ds)+i(8,^)}.
(2.3.2)
The pair (T, /i°) is unique.

66
MULTIVARIATE STABLE DISTRIBUTIONS
2.3
Definition 2.3.2 The vector X in Theorem 2.3.1 is said to have spectral
representation (T, £t°). The measure T is called the spectral measure of the a-stable
random vector X.
Warning. As we will see in Example 2.3.4, the components of the vector /x°, in
the case a = 1, are not equal to the shift parameters of the components Xu...,Xd
ofX.
The proof of Theorem 2.3.1 is sketched in the following remarks; for details,
see for example Kuelbs (1973).
Remarks
1. It follows easily from Definition 2.1.1 that a random vector X is a-stable if
it has characteristic function Oa(0).
2. Theorem 2.3.1 continues to hold if Rd is replaced by a separable Hilbert
space H with scalar product (•, •). For example, H can be £2, the space of
real sequences X = {x\,i2, ■ ■ ■} with ||x||2 = Y^kLi x\ < °° an(* scalar
product (x, y) = Y^kL\ xkVk- Theorem 2.3.1 for H — E2 can be used to
show that stable processes always have an integral representation.
3. To prove Theorem 2.3.1 in H (or in if = Rd), one shows that the
characteristic function of an infinitely divisible law in H with no Gaussian component
has the form
(see, e.g., Varadhan (1962) or Laha and Rohatgi (1979), p. 499.) Here, L
is the (unique) L6vy measure corresponding to that infinitely divisible law,
i.e., L is a a-finite measure, finite outside each neighborhood of zero, that
satisfies L({0}) = 0 and Jhx«<x ||x||2L(dx) < oo. Express x in terms of
its radial and angular components by setting x = rs, where r = ||x|| and
where s = x/ ||x|| belongs to the unit sphere S of H. Then, using a-stability,
prove that there is a measure a on S such that L(dx) = ;xW<ir cr(ds). The
characteristic function becomes
exp
\JsJo
*■<-.*> _ i _ i
r(s,x)
1+r2
dT-a(ds)+i(e, M°)
,-l+c*
Observe that the inner integral is the same as in the case H = M1 (see
(1.1.7)). After evaluating the inner integral, one obtains the final result,
formulated in (2.3.1) and (2.3.2) in the case H = Rd.

2.3 CHARACTERISTIC FUNCTION OF AN a-STABLE RANDOM VECTOR 67
Example 2.3.3 Suppose d = 1. Then Sj consists of the two points {-1} and
{1}, and the spectral measure T is concentrated on them. Write T(l) = r({l})
and T(-l) = T({-1}). If a ^ 1, 0>a(9) becomes
£exp{i6>X}=exp{-|6'l|0,(l-i sign (6*1) tan ^)r(l)
-|0(-1)|° (l - i sign (0{-l)) tan ™)r(-l) + ip°o}
= cxp{-|fl|0 [(HI) + r(-l)) - * sign (B)(T(l) - r(-l))tan™] +i^e).
It is the characteristic function of an a-stable random variable X ~ Sa(a, /3, p)
with parameters
a = (r(i) + r(-i))>/°, /g = rci) + re-1)' ^^
The skewness parameter {3 is zero if the spectral measure T is symmetric. An
analogous result holds for a = 1.
Example 2.3.4 Let X be an a-stable random vector with characteristic function
given in Theorem 2.3.1. We know by Theorem 2.1.2 that any linear combination
Yb = Ylk=\bkXk has an a-stable distribution 5a(crb,/3b,Mb)- To determine
the parameters <7b,/3b and /Xb characterizing that distribution, let 7 be any real
number and set 6 = 7b in (2.3.1) and (2.3.2). The resulting function of 7 is the
characteristic function of the random variable Yb, and one obtains
Eexp{i7Yb} = .Eexp{i(7blX)}
exp{- fSd |(7b,s)r(l - i [sign (7b,s)](tan ^)r(ds) + »(7b,/i0))},
exp{- JSd |7b,s|(l + i\ sign (7b,s)(ln |(7b,s)\)T{ds) + i(7b,M°))}>
' exp{-h|0[/Sd|(b,s)|ar(ds)
- * ( sign 7)(tan *») J^ |(b, s)\a sign (b, s)r(ds)] + i7(b, /x0)} ,
exp{-|7|[/sJ(b,s)|r(ds)
+ i |(sign7)/s<((b,s)ln|7(bls)|r(ds)] + fy(b,M0)},
for a ^ 1 and a = 1 respectively, so that
<rb = (J |(b,s)rr(ds)) , (2.3.3)
_ JsJ(b>s)rsign(b,8)r(ds)
A - /sj(b,8)|«r(ds) ' {23A)

68
MULTIVARIATE STABLE DISTRIBUTIONS
2.3
(b,M°) ifa^l,
/xb=<! (2.3.5)
(b,M0)-|/s<(b,8)ln|(b,8)|r(ds) ifo= 1.
By choosing suitable bs one can easily obtain the (marginal) distributions of
the components Xk, k = 1,2,... , d, of the vector X. It is easy to see, for example,
that the shift parameter of Xk is nk when a =£■ 1 but is fi°k - \ fs Sk In \sk \T(ds)
when a — 1.
Example 2.3.5 An a-stable random vector X = {X\, X2, ■ ■ ■, Xa) has
independent components if and only if its spectral measure T is discrete and concentrated
on the intersection of the axes with the sphere Sd. It is easy to verify this fact
by using the uniqueness of the spectral measure T. Suppose, for example, that
d = 2 and consider independent random variables Xi ~ Sa{cn,Pi, Mi)> i = 1,2.
Choose F concentrated on the points (1,0), (-1,0), (0,1) and (0,-1) by setting
T = a,5((l,0)) + a2<5((-l,0)) + a35((0,1)) + a46((0, -1)),
where a\, a2,03,04 are non-negative numbers and where 6(sq) assigns unit mass
to the point So- Because of the uniqueness of the spectral measure, it is enough
to verify that this choice of T is adequate. For convenience, assume a ^ 1. The
characteristic function (2.3.1) with the preceding choice of T is
*«(0i,fc)
= exp|- ^2 \8isi + 92s2\a(l -i sign (tfjsi + 92s2) tan— Jr(si,s2)
(sii«2)
+ t(0,/i?+fcjl§)}.
where £V S2) denotes summation over the four points (1,0), (-1,0), (0,1),
(0,-1). Equating this characteristic function to
Etxp{i{9lXl+92X2)}
= expj-cr?|0,|a (l -t/3, sign (9{) tan ™) + i9^x
- af\92\a (l - ip2 sign (&) tan ™) + i02/x2}
yields the system of equations
a\+a2 = of, facrf = a\ - a2, fi{ = /i°,
a-}+ 0,4 = a", ftc" = 03 — a4, /i2 = /x2,
whose solution
ql+A ql-A ql+ft „ . 1 ~ ft
a, = o-j —_—( a2 = ax — —( a3 - a2 —-—) a4 = a2 —^—

2.3 CHARACTERISTIC FUNCTION OF AN a-STABLE RANDOM VECTOR 69
provides the values of the weights in the expression of the spectral measure I\
The same result holds in the case a — 1.
Example 2.3.6 Let Yk ~ Sa(ak,Pk,Vk), k = 1,..., m, be independent random
variables and let A = {a,jk}, j = l,...,d, k = 1,...,m, be a real matrix.
Consider the random vector X = (X\ ,Xi,..., Xj) whose components are linear
combinations
m
Xj = 22 ajkYk, j = 1, • • •, d,
fe=l
of the YfcS. The vector X is a-stable with characteristic function
d to d
Eexp{»J2^Xj} = fiexpji^^^a^Yfc}
j=i fc=i j=\
m d
= Y[Eexj>{iJ2*i*3*Y><}- (2-3-6>
If a ^ 1, this is equal to
m d d
exp{-5ZcrfclIZ^aJfc|0'(1 -^fc^gn (5ZeJaJfe)tanT)
fe=l j=l
j=l
m a
k=i j=i
r d d d
exp{- / i5ZflJsJr(i-i«8n(5Z^)tanT)r(ds)+i^ej'x"}'
where
r-E
*=1
1+A a^ 2\o/2
<tf(£<&)° 5((^a,„Vw2—
Odfc
j=l
(E-=,^)1/2,""(E'=1^)1/2
))
+
l-fo-»
-s(E^r<(
-aiJc
j=i
and
(E-=1^)1/2'""(E-=.^)l/2
m
fc=i

70
MULTIVARIATE STABLE DISTRIBUTIONS
2.3
If a = 1, Relation (2.3.6) becomes
d
exp{~ ECTfcl £fli°J*l0 + * (~)& si§n (S*J°ifc) lnE^aJ'fc|)
m d
+*YlvkJ2eJaJk}
k=\ j=\
. d d d d
= exp{-jf I J28jSj\(l+i(-) sign(£6jSj) ln|£^Sj- r(ds))+» ]T0jAi°},
with T as above and
M° = 51 aJ* (^ - -akPk In 5Z anfc) •
fc=l n=l
Note that the spectral measure T of the a-stable random vector (X\,...,Xd)
is discrete and concentrated on m symmetric pairs of points (s^k\ — s^), k =
l,...,m, of Sd.
Conversely, an a-stable random vector (X\,..., Xd) with a discrete spectral
measure T concentrated on points (s^fc^, —s(fc)) as above can always be represented
as
m m
(Xu...,Xd) = C}J aikYk, • • • i ^ adkYk),
fc=i fc=i
where Yk ~ SQ(ak, Pk,P-k), k = 1,..., m, are independent random variables,
and A = {a.jk} is a matrix of real numbers. We leave the details to Exercise 2.9.
Example 2.3.6 and Exercise 2.9 yield:
Proposition 2.3.7 The spectral measure Td of an a-stable vector (X\,..., Xd)
is concentrated on a finite number of points on the unit circle Sd if and only if
(Xi ,■■■, Xd) can be expressed as a linear transformation of independent a-stable
random variables.
Note that the representation of the characteristic function of an a-stable
random vector in (2.3.1) and (2.3.2) involves integration over the unit sphere
Sd = {x: ||x|| = 1} of Md where || • || denotes the Euclidean norm. This is the
usual way of representing the joint characteristic function of an a-stable vector.
However, there are many other norms in M.d, and it is possible to use the unit
sphere relative to any one of them to represent the joint characteristic function of
an a-stable random vector.

2.3 CHARACTERISTIC FUNCTION OF AN a-STABLE RANDOM VECTOR 71
Specifically, let Sd be the unit sphere relative to the Euclidean norm and let T
be the corresponding spectral measure. Let || ■ || now be an arbitrary norm in Rd
and let Sd denote the corresponding unit sphere. We want to determine r||.||, the
finite Borel measure on Sd" that corresponds to spectral measure T on Sd.
Proposition 2.3.8 Let F\\.\\ be a finite Borel measure on Sd equivalent to T with
f|(.||(ds) = ||s||ar(ds) and let TH: Sd -» SJj'" be given byTs = s/||s||. Define
rll-li =f||-||o:riH|
and
( m(0) 1/096 1,
"P-lH m _ (2-3-7)
[ M(0)+M||.|| '/«=1,
where
(h-lOi = ~r / SJ ln llsllr(*0, j = 1,.. •, d.
* Jsd
Then the joint characteristic function <$>Q (9) q/7/ie a-stable random vector X in Md
is also given by (2.3.1) and (2.3.2) with (Sd, T, ^0)) replaced by (Sdl{l, TM ,n°H)
and(0,s) = J2dj=iejSj.
PROOF: A change of variables in (2.3.1) and (2.3.2) shows that they hold with
(Sd, T, n°) replaced by (Sd , IYy, Mn.ii)- The uniqueness of (r, fj.°) implies the
uniqueness of (ry. y, p ?,, ■,). I
We will usually work with the Euclidean norm in Rd but we shall encounter
other norms as well. The L°° norm ||x|| = maxi=1|...)(i \xi\, for example, is
a particularly natural one when studying the maxima of stable vectors. The
corresponding unit sphere is the boundary of the cube [— 1, \}d.
Suppose that the spectral measure and the shift vector of an a-stable
distribution on Rd are given. The following proposition shows how to derive formally
the spectral measure and the shift vector of any marginal distribution.
Proposition 2.3.9 lf(X\, X2,..., Xd) is an a-stable random vector with spectral
measure Td and shift n°d = (^, p%,..., fid), then (X\, X2, • ■ ■, Xn), n < d, is
an a-stable random vector with spectral measure Tn and shift /x°. The spectral
measure Tn can be expressed as
rn = h(rd) = rdoh-\
where
h:S'd = {(si,...,s„, sn+i,...,sd) eSd:s}-\ + s2n > 0}-+S„,

72
MULTIVARIATE STABLE DISTRIBUTIONS
2.4
h(si,...,sn, sn+i,...,Sd)
si
and
(E^I^)1/2'--"(E^1^)1/2/
a/2
4
fd(ds) = (£>?:) rd(ds), seS'd.
3=1
The jth component, j = 1,..., n, of the shift /x° equals
M? ~ i /s; *J ta(ELi 4)r«*(ds) i/« = 1-
Proof: We prove the result in the more complicated case a = 1. The proof in
the case a ^ 1 is similar. The joint characteristic function of X\, X2,..., Xn
satisfies
- In E expji £ BjXj } = - In E expji (Y^ 83X3 + Yl ° ' Xj) }
j=l j=l J=n+1
. n „ n n n
= / , l£*J*il(l +t- sign (£>^) ln|£0iaj-|)rd(ds) - »£^S
>< j=i
/*
= isJpJ(EL?4)1/2
(n
2
l-H-sign(X> '')
i=i
(£4) rd(ds)-i]>>M°
K£4)
= / iEe^i(1+^sign(E^))lniE^i)r"(dt)
j=l <i fc=l
2.4 Strictly stable and symmetric stable random
vectors
The expressions (2.3.1), (2.3.2) for the characteristic function OQ(0) of an
testable random vector involve the spectral measure T and the shift vector p°. Let

2.4 STRICTLY STABLE AND SYMMETRIC STABLE RANDOM VECTORS 73
us find conditions that T and /z° must satisfy for Q>a(8) to be the characteristic
function of a strictly (respectively, symmetric) a-stable random vector.
When is the characteristic function <i>a(#) that of a strictly a-stable random
vector X in Ed? We know by Theorem 2.1.5 that it is necessary and sufficient that
yb = Ylt=i bkXk be a strictly a-stable random variable for any b € Ed. When
a ^ 1, this amounts to requiring 0 = in, = (b, /j°) for any b, that is, /x° = 0.
When a = 1, the condition for strict stability is /3b = 0 for any b, i.e.,
0= / |(b,s)|sign((b,s))r(ds)= / (b,s)r(ds)
Jsd Jsd
for any b by (2.3.4), i.e., Js SfcT(ds) =0 for k = 1,2,... , d. There are no
conditions on jz°. Therefore,
Theorem 2.4.1 X is a strictly a-stable vector in Ed with 0 < a < 2 if and only
if
(a)a^ 1:
(b) a = 1:
/ sfcr(ds)=0 for k=\,2,...,d. (2.4.1)
Jsd
As a result, we obtain
Corollary 2.4.2 Suppose that X = (X\, X2, • ■ ■, Xj) is an a-stable vector in
Ed with 0 < a < 2. Then X is strictly stable if and only if all its components
Xk, k = l,...,d, are strictly stable random variables.
When is the characteristic function Oa (6) that of a symmetric a-stable random
vector X in Rd ? A necessary and sufficient condition is that /z° = 0 and T be
a symmetric measure on Sd (i.e., T(A) = T(-A) for any Borel set A of 5^). In
fact,
Theorem 2.4.3 X i'5 a symmetric a-stable vector in Ed with 0 < a < 2 if and
only if there exists a unique symmetric finite measure T on the unit sphere S^ such
that
£exp{i(0,X)}=:exp{- f \(0,a)\ar{ds)}. (2.4.2)
r is the spectral measure of the symmetric a-stable random vector X.

74 MULTIVARIATE STABLE DISTRIBUTIONS 2.4
PROOF: A necessary and sufficient condition for X to be symmetric is that its
characteristic function be real, i.e., (2.4.2) holds. Now define f by f(A) =
j(r(i4) + T(-A)) for all Borel sets A of S* and observe that f is a symmetric
measure. Since (2.4.2) also holds if we replace T by T, we must have T — T by
unicity (Theorem 2.3.1) and hence T is symmetric. ■
A symmetric a-stable distribution in Rd is denoted SaS. We also say that
the vector X = (X\,X2, ■ ■ ■ ,-Xd) is SaS in M.d or that the random variables
X\, X2,... ,Xd are jointly SaS. Figure 2.1 illustrates a SaS distribution with
a = 1.5 and spectral measure T with mass 0.2 at 0° and 180° and mass 0.8 at
90° and 270°. It displays the probability density function and the integral of that
density from the point (—2, —2).
Example 2.4.4 If d = 1, one has Si = {-1,1}, r({l}) = r({-l}) and hence
an a-stable SaS random variable X has distribution Sa(cT, 0,0) with
a = (f \sfnds))i/a = (2rai})y/a.
Remarks
1. Can a strictly 1 -stable random vector X be made symmetric by shifting? To
answer this question, we must consider two cases. Suppose, firstly, that X
has a symmetric spectral measure T. Then shifting X by pi transforms X into
a symmetric random vector X — //. Suppose, now, that the spectral measure
r of X is not symmetric. This can never happen when d = 1 because the
condition for strict stability 0 = fs s T(ds) = r({l})-r({-l}) requires
r to be symmetric. When d > 1, the conditions for strict stability are
fs SkT(ds) = 0 for fc = 1,2,..., d. They can be satisfied by a non-
symmetric spectral measure T, for example, by
m = *((1,0)) + S((0,1)) + y/2S((-j=, ~^))
if d = 2. If X is a strictly 1-stable random vector with a non-symmetric T,
then X — fj. will not be symmetric, because shifting X has no effect on T:
the vector X — /x and X have the same non-symmetric spectral measure T.
Hence
Corollary 2.4.5 Not every strictly l-stable random vector in Rd with d > 1
can be made symmetric by shifting.

2.4 STRICTLY STABLE AND SYMMETRIC STABLE RANDOM VECTORS 75
Figure 2.1: 3-D plots of an a- density function and of the integral of that density from
the point (-2, —2); here a is 1.5, and the spectral measure has mass 0.2 at 0° and 180°
and mass 0.8 at 90° and 270°.

76 MULTIVARIATE STABLE DISTRIBUTIONS 2.4
2. Unlike Corollary 2.4.2, the symmetry of an a-stable random vector
cannot be regarded as a componentwise property. There are non-
symmetric a-stable random vectors (Xi, X2,..., Xd) whose components
X\, X2, ■ ■ •, Xd are all symmetric.
For example, let X\,X2,Xz be i.i.d. Si(1,1,0) and consider the vector
Y = (Yi,y2), where
The random variables Yj, j = 1,2, are jointly 1-stable and they are
symmetric because they have a skewness parameter equal to zero by (2.3.4)
and a shift parameter equal to zero by (2.3.5). However, the random
vector Y is not symmetric because, by (2.3.5), the linear combination
0i Yi + 62Y2 = 9xXi + (62 - 0\)X2 - 62X1 has a non-zero shift parameter
if 0i and 62 are non-zero and unequal.
To illustrate the point when a j4 1, suppose that X = {X\,X%) is an
a-stable random vector with spectral measure
r - 6((i,o)) + «((o, 1)) + 2a/26((-^=, ~)),
and shift /x° = 0. Then Xl and X2 are symmetric by (2.3.4) and (2.3.5) but
X is not symmetric because its spectral measure is not symmetric.
3. Expression (2.4.2) holds also in the Gaussian case a = 2, but then T is not
unique. Consider the following example:
Let X\, X2 be correlated N(0,1) random variables with p = EX\ X2. The
characteristic function of X = (X\, X%) is
£exP{i(0,x, + e2x2)} = exp{-i(0? + 2pexe2 + e22)].
There are many symmetric measures T on S2 such that
1(0? + 2p0,02 + 9l) = / (s,0! + s202)2r(ds).
2 JS2
In particular, any T such that
r = Y, J°*{fi((6*. (! ~ bfc)1/2)) + 5((~6^ ~(l - 6£)1/2))}'

2.5
SUB-GAUSSIAN RANDOM VECTORS
77
with afc > 0,0 < bk < 1 and
£>62 = 1, £afc&fc(l -b\y'2 = p, 5>*U "#) = 1,
k k
satisfy this identity, for instance,
r=^{(l-p)[6((l,0)) + ,5((-l)0)) + 5((0>l)) + 6((0,-l))]
+"K(7!^))^((-7!.-^)]}.
or1
r=^(l-P2)'/2[«((0,l)) + 5((0,-l))]
+ ^1+^K((r+?^'(r+^))
+ 6lV(l+p2)l/2'(l + p2)I/2j)J}-
The spectral measure is not a useful concept in the Gaussian case a = 2,
because it is not unique.2
4. Let X\,X2,---,Xd be jointly SaS with spectral measure I^. Then
X\,X2, ■ • ■ ,Xn, n < d, are jointly SqS with spectral measure Tn
defined in Proposition 2.3.9.
5. Unlike the Gaussian case a = 2, the distribution of a SaS random vector
on Ed, d > 3, is not determined by its two-dimensional marginals. (See
Exercise 2.13.)
2.5 Sub-Gaussian random vectors
We have seen in Section 1.3 that the random variable X — AXI2G is SaS, a < 2,
if G ~ N(Q, a2) and A is an a/2-stable random variable totally skewed to the
right and independent of G. This result extends to random vectors X as follows.
Choose
i4~SQ/2((cos^)2/a,l,0) (2.5.1)
1 This second T can be readily obtained from the characteristic function of (X\, X2) by representing
X2 us pXi + (1 — p2)'/2Y where Y is a iV(0, 1) random variable independent of X|.
2 The spectral measure T should not be confused with a measure F, also sometimes called spectral
measure, which appears in the context of a stationary Gaussian process {Xt}. The measure F is
denned by EX0Xt = f™ exp{i\t}dF{\).

78 MULTIVARIATE STABLE DISTRIBUTIONS 2.5
with a < 2, so that its Laplace transform is
Ee-<A = e~ian , 7 > 0, (2.5.2)
by Proposition 1.2.12. Let
G = (C7i,G2,...,Gd)
be a zero mean Gaussian vector in Rd independent of A. Then the random vector
X = (i4|/2G,,i4,/2G2l...lA|/2Gd) (2.5.3)
has a SaS distribution in Rd because, for any real numbers 61,62,... ,bd, the linear
combination J2k=i hAi/2Gk = Al/2 £^=1 bkGk is a SaS random variable
(Proposition 1.3.1) and hence, by Theorem 2.1.5, X is SaS.
Definition 2.5.1 Any vector X distributed as in (2.5.3) is called a sub-Gaussian
SaS random vector in Rd with underlying Gaussian vector G.3 It is also said to
be subordinated to G.
The characteristic function of a sub-Gaussian random vector has the following
special structure:
Proposition 2.5.2 The sub-Gaussian symmetric a-stable random vector X
defined in (2.5.3) has characteristic function
d 1 d d
Ecx?[iY,hXk}=cxV{~\-YJY/eiejRij\a/2}, (2.5.4)
fc=l i=l j = l
where fly = EGiGj, i,j = \,...,d, are the covariances of the underlying
Gaussian random vector (G\, G2,..., Gd).
PROOF: By conditioning on A we obtain
d d
E^V{iJ2&kXk} = EE[exV{iAl/2Y,OkGk}\A]
fc=i fc=i
. d d
= .Eexpj-A-^^^Mi}
- exP{-|ii:]CiW;n/2},
since Esxp{-"/A} = exp{-7Q/2}, 7 > 0. I
3Some authors use the term sub-Gaussian to refer to a class of stochastic processes with exponential
moments. We do not use the term sub-Gaussian in that context.

2.5
SUB-GAUSSIAN RANDOM VECTORS
79
Example 2.5.3 The characteristic function of a multivariate Cauchy distribution
in Rd is
4>{9) = exp{-(0TI0)'/2 + i(9,n0)},
where T denotes a transpose and X is a d x d positive-definite symmetric
matrix. The multivariate Cauchy distribution is thus a shifted SIS sub-Gaussian
distribution. Its density function is
f (v\ _ Ll I -y f= TOd
/W [l + (x-M0)Z-i(x-M°)](d+1)/2 '
where |£| denotes the determinant of 2 and
1 „/d+l'
c =
7r(d+i)/2
(See Press (1972).)
m
Corollary 2.5.4 Let X he sub-Gaussian SaS with underlying Gaussian vector
G. Then there is a one-to-one correspondence between the probability distribution
of G and that of X.
PROOF: Let G' be a mean zero Gaussian vector on R.d with covariances
R'i:j, i,j = 1,... ,d. We must show that if X' = AX,2G' has the same
distribution as X = A^2G, then G' has the same distribution as G. By Proposition
2.5.2, X' = X implies
d d d d
Yl ]C eieJR'ij = Yl zZ ®ieJRiJ
t=l j = \ i=l j=l
for any real 9\,Q2,... ,8d- This identity between two polynomials in the ^s
implies equality of the coefficients, i.e.,
R'ij =Rij, i,j = l,...,d.
Hence G' = G.
We noted in Theorem 2.4.3 that a SaS random vector has a symmetric spectral
measure T. This merely means that T assigns equal weights to antipodal sets on
the sphere Sd. We now want to characterize the SaS random vectors in Ed that
have a uniform spectral measure T, i.e., for which there is a constant C > 0 such
that T(B) = C\B\ for any Borel set B of Sd.
Proposition 2.5.5 Let X be a SaS, a < 2, random vector in Ed. Then the
following three statements are equivalent:

80 MULTIVARIATE STABLE DISTRIBUTIONS 2.5
(a) X is sub-Gaussian with an underlying Gaussian vector having i.i.d.
N(0, a2) components.
(b) The characteristic function o/X is of the form
d
£exp{i]T0fcXfc} = exp{-2-«/V|0n,
fc=i
i.e. it depends only on the magnitude of the vector 6 = (9U... ,9d).
(c) The spectral measure o/X is uniform.
PROOF: (a) is equivalent to (b) because if X = (X\,X2,...,Xd) is a
sub-Gaussian SctS random vector whose underlying vector has components
G\, G2,..., Gd i.i.d. N(0, a2), then by Proposition 2.5.2, X has characteristic
function
k=\ k=\
To see that (c) => (b), suppose that X is SctS with uniform spectral measure
r on Sd- Then X has characteristic function
d
Eexp{iJ20kXk}=exp{-f(91,62,...,ed)} (2.5.6)
with
r d
M,62,...,ed)= / \Y,Oksk\aT(ds).
Js* fe=l
Since T is uniform, / depends only on the Euclidean norm |0| of the vector
0 = (0i,#2, ■ • • ,0d) € Md. Moreover,
f(cBi,c02,...,cBd) = caf'(fl,,^,..., ed)
for any c > 0. This means that
m,e2,...,ed) = c(jr,e2y/2
for some C > 0. Choosing 0 such that C = 2~al2oa, we may equate the
characteristic functions (2.5.5) and (2.5.6). This proves (c)=Kb). The converse
follows from the uniqueness of the spectral measure T. I

2.5
SUB-GAUSSIAN RANDOM VECTORS
81
Example 2.5.6 Suppose that X = (X\, X2) has characteristic function
0(0,, 02) = e-CTV^f+«l, _oo < 0,, 02 < oo.
Then X is a sub-Gaussian SIS random vector with a uniform spectral measure.
The density function of X is
f(xux2) = . 2 —r-—jttt,, -co < xux2 <oo.
27r(xf +15 + a1)*!2
This is the isotropic Cauchy density in R2.
The components G\, G2,..., Gd of the underlying Gausssian vector G are in
general not i.i.d. but, being Gaussian, they can always be expressed as a linear
combination of i.i.d. N(0,1) random variables. Therefore,
Proposition 2.5.7 Let Z be a SaS sub-Gaussian random vector in Rd with
underlying Gaussian vector having i.i.d. N(0,1) components. Then for any SaS
sub-Gaussian random vector X in M.d, there is a lower-triangular dx d matrix A
such that
X = AZ.
The matrix A is of full rank if the components o/X are linearly independent.
The proof is straightforward and is left as an exercise (Exercise 2.16).
What is the general form of the spectral measure T of a sub-Gaussian SaS
random vector X? We noted in Proposition 2.5.5 that it is uniform if the underlying
Gaussian vector G has i.i.d. components G\,G2y..., Gd- This fact will enable us
to show that the spectral measure T of a general sub-Gaussian vector is a transform
of the uniform measure on Sd-
Let r0 be the uniform measure on Sd satisfying
Js* j=\ j=l
and let X = A1/2G be a sub-Gaussian SaS random vector in Rd with spectral
measure T. Let E = ((.Ry, i,j - 1,... ,d)) be the covariance matrix of the
underlying Gaussian vector G. Since Z is non-negative definite, there is a (lower-
triangular) d x d matrix A such that
Z = AAT,
(2.5.8)

82 MULTIVARIATE STABLE DISTRIBUTIONS 2.5
where T denotes a transpose. Our goal is to find a transformation
h: Sd—* Sd
such that
r = Mro) = r0oft-1.
We do this in three steps:
(1) The measure T0 can be regarded as a measure on Kd concentrated on the
unit sphere Sd- In this first step, we let
. hi : Rd -* M.d
be the transformation induced by the matrix A and
mi = /ii(r0) = r0o/i~'
be the induced measure on Ed.
(2) In the second step we transform m, into
d ,2
m2(ds) = (^s|J mi(ds),
which is also a measure on Rd. Write mz = /i2(mi).
(3) In the third step, we map Rd\{0} into the unit sphere Sd by applying the
transformation
h3 : Rd\{0} -> Sd,
where
Ms"5! Sd) = (^W* ed^)'
and we define
7713 = h-$(m,2).
We then have m3 = /i(r0) where
h = h3oh2ohi. (2.5.9)
We claim that 7713 is, in fact, T, the spectral measure of the sub-Gaussian random
vector X.
Proposition 2.5.8 The spectral measure T of a sub-Gaussian SaS random vector
in Rd has the form
r = h(r0),
where To is the uniform measure on Sd defined in {25.7) and where h'.Sd^Sd
is defined in (2.5.9).

2.5
SUB-GAUSSIAN RANDOM VECTORS
83
PROOF: By Theorem 2.4.3, we have
r d
£exp{i(0,X)} = exp{- / |]TejSj\aT{ds)\.
Since the spectral measure is unique, it is enough to show that we can replace T
by m3 = h(To). Now,
Js* J=1 Mm fri (ELi4)1/2
r d r d
-/ |]>]0jSj|Qm,(ds) = / iV^s.rm.Cds)
= /" |(0,As)|aro(ds)= / |(e,As)|ar0(d8)
JRd JSd
= / |(AT0!S)|Qro(ds)=(i^(Ar^)a/2 (using (2.5.7))
= (I(A^A^))a/2-(I(a,AA^))0/2
which, by (2.5.4), is the exponent of the characteristic function of X. This proves
that r = m3 = h{T0). I
Corollary 2.5.9 Let X = (X], Xz) be a sub-Gaussian SaS random vector in R2
with underlying Gaussian vector G = (G\, G2)- Xnen Jf i and X2 are dependent
unless G\ = 0 a.s. or G2 = 0 a.s.
PROOF: As before, we let r = h^mz), vnz = /i2(mt) and mi = /ii(r"o).
Suppose, to the contrary, that G\ ^ 0 a.s., Gz ^ 0 a.s. and that X| and X2 are
independent. Then the spectral measure T =z= 0 of X is concentrated on the points
(0,1), (0, -1), (1,0), (-1,0) of the unit circle S2 (see Example 2.3.5). Therefore
m2 is concentrated on the axes of M2, and so is mi. Since mi = h[ (Tq), where To
is the uniform measure on S2, we can find two linearly independent vectors s^
and s<2) on Si such that h 1 (s( l >) and h 1 (s(2)) are both concentrated on either the
"x axis" or the "y axis" of R2. Suppose without loss of generality that they are
concentrated on the "x axis." Since h\ mapes two linearly independent vectors
into that subspace, it maps the whole of R2 into that subspace. This means that

84 MULTIVARIATE STABLE DISTRIBUTIONS 2.6
the matrix A in (2.5.8) has the form
so
^(")C;)=Cw).
and hence Var G\ = 0. This implies G\ = 0 a.s., contradicting the hypothesis.
The proof is now complete. I
Not all symmetric a-stable random vectors are sub-Gaussian. For example,
by Corollary 2.5.9, a vector with independent non-zero components cannot be
sub-Gaussian. We will see in Section 6.7 that the components of a sub-Gaussian
SaS random vector are, in fact, strongly dependent.
2.6 Complex SaS random variables
Although we are mainly interested in real-valued random elements, we will later
encounter an important class of real-valued stable processes (the so-called "har-
monizable processes"), which are typically defined in terms of complex SaS
random variables. Let X\ and X2 be real random variables defined on the same
probability space. It is the joint distribution of X\ and X2 that characterizes the
complex random variable X = X\ + iX2.
Definition 2.6.1 A complex random variable X = X\ + iX2 is called symmetric
a-stable (SaS ) if the random vector (Xi, X2) is SaS in IS.2.
Definition 2.6.2 A complex SaS random variable X = X\ + iX2 is rotationally
invariant (or isotropic) if
e**X = X
foranyc/>€ [0,2tt).
Warning. Some authors use the term complex SaS random variables to mean
random variables that are both complex SaS and isotropic.
Since (complex) isotropic SaS random variables play an important role in the
context of so-called "harmonizable processes," we shall examine in greater detail
the structure of the joint distribution of X\ and X2 in the isotropic case. Isotropy
is a strong condition. For any </> e [0,27r), one must have
(cos <j> + i sin <j>)(X 1 + iX2) = (X, + iX2),

2.6
COMPLEX SaS RANDOM VARIABLES
85
i.e.,
(Xi cos <j> - X2 sin <j>, X2 cos (j> + X\ sin (j>) = {X\, X2)
or
Rj, X = X,
where
d _ ( cos<£ sin^>
* y — sin 4> cos </>
is the matrix corresponding to the rotation by an angle <f>, T denotes transpose,
andX = (X,,X2).
If a = 2, the isotropy condition e^X = X is satisfied by X = X1 + iX2
with Xi, X2 i.i.d. AT(0, ct2), since
£(Xi cos </> - X2 sin <^)(X2 cos <j> + Xx sin (j>) = 0 = JSXi X2,
E(Xi cos <£ - X2 sin <£)2 = E(X2 cos 0 + Xx sin <?!>)2 = a2,
and, in fact, one can easily verify that if X\, X2 are not i.i.d., then X = X\+ iX2
is not isotropic (see Exercise 2.18.) We will see below that when a < 2, the real
and imaginary parts of an isotropic SaS random variable are dependent.
Let us suppose a < 2. The following theorem shows that isotropy implies a
very special spectral measure for X = {X\, X2).
Theorem 2.6.3 Let a < 1. A complex SaS random variable X — X\ + 1X2 is
isotropic if and only if (X\, X2) has a uniform spectral measure.
PROOF: Let 9 = (6\, 02). The characteristic function of X = (X\, X2),
£exp{z(0,X)} = exp{-y |(0,s)|Qr(ds)}
must be equal to the characteristic function of -RjX, i.e., to
£exp{i(0,.R£X)} = Eexp{»(fl*0,X)}
= exp{- f \(R^,s)\aT(ds)}
= exp{-J |(0,J^s)rr(ds)}
= exp{- f \{e,t)\aT^dt)}
where we set t = R^s and
r*(B) = r((jjJ)-,(B)) = r(j^(B))

86
MULTIVARIATE STABLE DISTRIBUTIONS
2.6
for any Borel set B of 82- Because of the uniqueness of the spectral measure, this
occurs if and only if
r = rv,v4>e[o,27r),
that is, if and only if T is a uniform measure on 52. 1
In view of Proposition 2.5.5, we have
Corollary 2.6.4 Let a < 2. A complex SaS random variable X = X\ + iX2
is isotropic if and only if there are two i.i.d. zero mean normal random variables
G\ and G% and a random variable A ~ SQ/2((cos7ra/4)2/Q, 1,0) independent
of{G\, G2) such that (XuX2) is sub-Gaussian with underlying vector (Gi, G2).
Thus {X\,X2) = {A^2Gi,A^2G2). In other words, every complex isotropic
SaS random variable with a < 2 is of the form
X = A1/2(G]+iG2).
Corollary 2.5.9 implies that the real and imaginary parts of X are always dependent
(unless G\ and G7 are degenerate).
Corollary 2.6.5 Let a < 2 and suppose that the complex SaS random variable
X — X\ + 1X2 is isotropic. Then
Eei{elxl+02x2) _ e-cor(s2)|0r
where T is the spectral measure of{X\, X2) and
1 /*27r
°°= 2tt / lC0S^Q£^-
PROOF: The result follows from Corollary 2.6.4 and Proposition 2.5.5, or directly
by noting that the characteristic function of X = (X\, X2) is
£exp{i(0,X)} = exp{-| |(0,s)rr(ds)},
where T is a uniform measure on 52- Letting k, and 4> denote, respectively, the
arguments of the vectors 6 and s, we have
£exP{i(0,X)} = e*v[-W j\cos{K-4>)\aT-^^}
= exp{-<*r(Sz)|0|°}. ■

2.7
COVARIATION
87
This shows that the characteristic function of X = (Xx, X2) depends only on
the modulus of 6.
When a = 2, the necessity part of Theorem 2.6.3 fails because T is no longer
unique. In fact, we noted at the beginning of this section that when a— 2, the real
and imaginary parts of X are independent. This is a major difference between the
cases a = 2 and a < 2.
2.7 Covariation
The covariance function is an extremely powerful tool in the study of Gaussian
random elements, but it is not defined when a < 2. The covariation is designed
to replace the covariance when 1 < a < 2. Unfortunately, it is not as powerful
a tool, because it lacks some of the desirable properties of the covariance. It is,
however, a useful quantity and it appears naturally in many settings, for example,
in the context of linear regression.
In this section, we define the covariation of two jointly SaS random variables
with 1 < q < 2, and compare its properties with those of the covariance. We start
with the definition of "signed power." Let a and p be real numbers. The signed
power a<p> equals
a|psigna=< (2.7.1)
For example, {-2)<l'2> = -V2 and (-2)<2> = -4. In general, a<1> =
a, a<2> £ a2 and {ab)<P> = a<P>b<P>.
Definition 2.7.1 Let Xx and X2 be jointly SaS with a > 1 and let T be the
spectral measure of the random vector (A'i, X2). The covariation of X\ on4 X2
is the real number
[XuX2}a= f s,s<a-l>T(ds). (2.7.2)
JSi
Example 2.7.2 If a = 2, the characteristic function of {X\, X2) is
£exp{i(0,Xi + 62X2)} = exp{- f (0is, + 62s2)2T{ds)j
= exp{-(02 f s2lT{ds)+2eld2[Xl,Xz}2 + elJ slnds))}.
4As we shall see, the covariation is not symmetric in its arguments. Nevertheless, [X|, X2]a is
often called "the covariation of Xj and X2"

88 MULTIVARIATE STABLE DISTRIBUTIONS 2.7
On the other hand,
Eexp{i{6iXi+e2X2)}
= exp{--(0? VarX, + 20,02 Cov(X,, X2) + 6\ VarX2)l.
Setting first 9\ = 0 and then 62 = 0 yields
^r(ds) = i VarX,-, j = 1,2,
and hence
[XuX2}2=1-Cov(XuX2).
The definition of the covariation [XuX2]a involves the spectral measure T
which appears in the spectral representation of the characteristic function of the
vector (Xi, X2). We will see in the next chapter that the covariation [X\, X2]a can
also be expressed in terms of the so-called "integral representation" of (X\, X2).
It is possible, however, to give a definition of the covariation which is not
related to representations and which can be used when the representations are
difficult to obtain explicitly. To state this second definition, proceed as follows.
Let (X\, X2) be jointly SaS, 1 < a < 2, and consider the SaS random
variable
Y = 8\X\ + #2-^2-
where 9\ and 82 are real numbers. Let a(6i,02) be the scale parameter of the
random variable Y.
Definition 2.7.3 (equivalent to Definition 2.7.1). The covariation \X\,X2)a of
(X,,X2)is
\daa(9ue2
[Ai,A2]Q = -
5(9,
(2.7.3)
e,=o,e2=i
To prove that this definition is equivalent to Definition 2.7.1, let T denote the
spectral measure of (X\, X2). Then by (2.3.3),
<Ja{0i,02)= f |0,s,+02S2rr(ds).
Js,
>s2
It is easy to check that the finiteness of T and the fact that a > 1 imply that
= a [ s,(0,
Substituting 0j = 0 and 82 = 1 yields (2.7.3). This shows that Definitions 2.7.1
and 2.7.3 are equivalent.

2.7
COVARIATION
89
Example2.7.4 Let (Gi,G2,... ,Gn) be mean zero jointly Gaussian random
variables with covariance Rij = EGiGj, i,j=l,...,n. Fix 1 < a < 2
and let A ~ Sa/2 ((cos ^ff>a, 1,0) be independent of (G,, G2, • •., G„).
The random vector X = (X\, X2,..., Xn) with
Xk=Al*Gk, fc=l,2,...,n,
is sub-Gaussian (see Section 2.5). Let us compute the covariations
[Xi,Xj}a, i,j = 1,... ,n, of the components of X. Since X has
characteristic function
n n n
EexV{iJ28kXk} = expj^-^l^r^^iy^2}
fc=i i=i j=i
(Proposition 2.5.2), we see that the scale parameter a{9i,9j) ofY = 9iXi + 9jXj,
i,j ~ 1,... ,n, satisfies
aa{9t, 9j) = 2~Q/2 (fiRii + MiAjRij + OJRjj)^2 .
Using (2.7.3), we see that
(Observe that here pfj, Xj]a = [Xj, Xi]a if i?™ = Rjj-)
As an application of (2.7.3), we have
Lemma 2.7.5 Let X = (X\, X2, ■■■, Xn) be a SaS random vector with a > 1
and spectral measure Tx, and let Y = Y^k=i akXk and Z = Y^k=\ ^kXk. Then
\Y>z}o=Js (x>**) (i>*k) rx(ds)-
PROOF: The scale parameter a{9\ , 62) of the linear combination 9\Y + 92Z is
because
(f \Y/(9[ak + 92bk)sk\arx(dsj)
exp{-ffa(fl,,fl2)} = Eexp{i{9[Y + 92Z)}
n
= Eexp|i53(fliafc+fl2bfc)^fe}
fc=i
= exp{- f |J>iafc + 02&fc)afc|arx(ds)}.
•'S" fc=l

90 MULTIVARIATE STABLE DISTRIBUTIONS 2.7
Applying (2.7.3) yields the desired result. I
The following corollary expresses the covariation of two components of a
random vector in terms of the spectral measure of the vector.
Corollary 2.7.6 Under the conditions of Lemma 2.7.5,
[XuX2]a = / slS<a-l>T^ds)
and
[xl,xl]a= [ \8l\arx(ds) = *%t
Js„
where ax{ is the scale parameter of the SaS random variable Xy.
We now list a number of properties of the covariation. We suppose throughout
that 1 < a < 2. The first two properties follow directly from Lemma 2.7.5.
Property 2.7.7 (Additivity in the first argument). Let (XUX2,Y) be jointly
SaS. Then
[x1 + x2,y]Q = [x1)Y]Q + [x2,y]Q.
Property 2.7.8 (Scaling). Let (X, Y) be jointly SaS and let a and b be real
numbers. Then
[aX,bY]a = ab^-^iX^Y}*.
The preceding properties point to important differences between covariation
and covariance. For example.
Corollary 2.7.9 Although the covariation is linear in its first argument, it is in
general not linear in its second argument.
The linearity in the first argument,
[aiX1+a2X2,Y}a = cn{XuY}a + a2lX2,Y}a,
follows from the preceding properties. Observe that the covariation may not even
be additive in its second argument, i.e.,
[X, y, + Y2}a ± [X, Yt]a + [X, Y2]a,
since one has
[X,X + X}a = 2a-l[X,X}a ± 2{X,X}Q = [X,X]a + \X,X]a
for a < 2 and [X, X]a J= 0. (By Corollary 2.7.6, this last requirement is
equivalent to X ^ 0 a.s.)

2.7
COVARIATION
91
Corollary 2.7.10 The covariation is, in general, not symmetric in its arguments.
To verify that, in general,
[X,Y]a^[Y,X\a,
let q < 2, [X, X] 7^ 0 and let a and b be two different non-zero real numbers.
Then
[aX,bX]a = ab<a-l>[X,X]a ^ a<a-l>b[X,X}a = [bX,aX]a.
The next property states that like the covariance, the covariation is zero when
its arguments are independent.
Property 2.7.11 IfX and Y are jointly SaS and independent, then
[X,Y)Q=0.
PROOF: We saw in Example 2.3.5 that the independence of X\ and X2
implies that the spectral measure T of (Xi, X2) must be concentrated on the points
(1,0),(-1,0),(0,1) and (0,-1) of S2. Wethenhave
[X,Y]a= [ Sls<a-l>T{ds) = 0
Js2
since the support of T is such that either s\ or S2 is zero. I
Example 2.7.12 Unlike the Gaussian case a = 2, it is possible in the case
1 < a < 2 to have [X, Y\a = 0 with dependent X and Y. For example,
let X = A1/2Gi, Y = Al/2G2 be jointly sub-Gaussian, with GX,G2 i.i.d.
(non-degenerate) standard normal random variables. It follows from (2.7.4) that
[X, Y\a = [Y, X]a = 0. But X and Y cannot be independent by Corollary 2.5.9.
We observed that, in general,
[X,Y{ + Y2}a^[X,Yl)a + [X,Y2}a.
We shall now prove that equality holds when Y{ and Y2 are independent. This
property turns out to be very useful in the context of stochastic integrals. We start
with the following lemmas.
Lemma 2.7.13 Let u and v be real numbers. 7/0 < p < 1, then
|u + u|p < Mp + Mp (2.7.5)
with equality, ifp< 1, if and only if either u = 0orv = 0.If\ <p<oo, then
\u + v\P<2p-l(\u\p + \v\p). (2.7.6)

92 MULTIVARIATE STABLE DISTRIBUTIONS 2.7
Proof: Since (\u + v\)p < (\u\ + \v\)p, we may suppose u > 0 and v > 0.
If 0 < p < 1, then, for fixed v > 0 and for any u > 0, we have #„(u) :=
up + vp - (u + v)p > 0 with equality only at u = 0, because gv(Q) = 0 and
g'v(u) > 0 for u > 0. If 1 < p < oo, then, by Jensen's inequality
(u + v)p = 2p (~^) <2pl-(up + vp). I
Lemma 2.7.14 Let (E, £, m) be an arbitrary a-finite measure space, and let
fi : E -+ K, i = 1,2, be two functions in La(E, £, m).
(1) If0<a< 1, then either of the relations
[ \fs(x) + f2(x)\arn(dx) = f IMxWmidx) + f \f2(x)\am(dx)
Je Je Je
or
[ \fi(x) ~ f2(x)\am(dx) = f \fi{x)\am{dx)+ [ \f2(x)\am(dx)
Je Je Je
implies
f\(x)f2{x) = 0 m-a.e.
(2) If 1 < a < 2, then
[ \fi(x) + f2(x)\am(dx) = / \fi(x) - f2(x)\am(dx)
■ Je Je
= I \fx{x)\am{dx) + [ \f2(x)\am(dx)
Je Je
implies
fi{x)f2{x) =0 m-a.e.
PROOF: (1) Suppose 0 < a < 1. By Lemma 2.7.13,
/ \fi(x) + f2(x)\am(dx) < f |/,(i)rm(dx)+ / \f2(x)\am(dx), (2.7.7)
Je Je Je
with equality if and only if f\ (x)/2(x) = 0 m-a.e. Replacing f2 by -f2 in (2.7.7)
completes the proof of the first part of the lemma.
(2) Suppose, now, 1 < a < 2. Adding the equations, we obtain
f |/,(x) + f2(x)\am(dx)+ f |/i(x) - f2(x)\am(dx)
Je Je
= 2 (/ \fi(x)\am(dx) + J \f2(x)\am(dx)\ . (2.7.8)

2.7
COVARIATION
93
We prove that (2.7.8) implies the conclusion of the proposition. Since 0 < a < 2,
the function r(u) = iW2, u > 0 is concave. Therefore, for fixed x € E,
|/,(x) + /2(s)|Q + |/l(z)-/2(z)|a
= U/i(s) + /2(x)]2)Q/2 + ([/,(*) - /2{x)]2)Q/2
= r([/,<x) + /2(x)]2) + r([/,(x) - /2(x)]2)
= 2r([/,(x) + /2(x)]2) + r([/,(x) - /2(x)]2)
^^/.(^/.Mfjl/.W-Mj (concavity) (2.7.9)
= 2r(/?(x) + /|(x))
= 2(/f(x) + /2(x))a/2
< 2 ((/2(x))a/2 + (/22(x))Q/2) (exponent a/2 < 1)
= 2 (|/,(x)r+ |/2(i)|a)
with equality in the preceding relations equivalent either to /i(x) = 0 or to
/2(x) = 0. (To verify this, notice that since r is strictly concave, the first
inequality in (2.7.9) becomes an equality if and only if (/i + /2)2 = (/i - /2)2,
that is, /i/2 = 0. For the second inequality, use Lemma 2.7.13.)
Now (2.7.9) implies that the left-hand side of (2.7.8) is always less than or
equal to the right-hand side of (2.7.8) and, if they are equal, then necessarily
(2.7.9) holds with equality for m-almost any x € E. Therefore, for m-almost any
x e E, we have f\ (x)/2(x) = 0. I
We are now in position to prove
Property 2.7.15 Let (X, Yi, Yi) be jointly SaS, a > 1, with Y{ andY2
independent. Then
[X,Yl+Y2]a = {X,Y{\a + [X,Y2}a.
Proof: Let T be the spectral measure of (X, Yx, Y>). The characteristic function
of (y,,y2) is
£expfi(0iY, + 92Y2)} = exp|- f |0 • s, + 0is2 + 02s3rr(ds)},
but, by independence of Y\ and Y2, it is also equal to
£exp{i0iYi}£exp{i6>2Y2}
= exp{- / |0-s,+elS2+0-s3|Qr(ds)}exp{- J |0-s,+0-s2+e2s3|ar(ds)}.

94 ' MULTIVARIATE STABLE DISTRIBUTIONS 2.7
Therefore
/ |0i*2 + 02a3rr(ds) = |0,|a [ |s2|°r(ds) + \e2\a [ Mar(ds)
JS} JS3 JSi
holds for all Q\ and 02. By Lemma 2.7.14, this implies
s2S3 = 0 T a.e.
and hence,
f s,(s2 + s3)<a-1>r(ds)= f s1s2<Q-1>r(ds)+ f s,s3<Q-1>r(ds).
J Sj J S3 J Si
Using Lemma 2.7.5, we obtain
[x,y, + y2]Q = [x,y,]a + [x,y2]a. ■
The covariation [X, Y]a is related to the joint moment EXY<p_'>.
Lemma 2.7.16 Let (X, Y) be SaS with a > 1. Then for alll<p<a,
EXY<p~]> _ [X,Y}a
e\y\p ~ ||y||s '
vWiere || V ||Q denotes the scale parameter ofY.
Proof: Let Yq be a SaS random variable with scale parameter \\Yo\\a = 1, and
let T be the spectral measure of the vector (X, Y). Since the random variable
AX+ y has scale parameter ||AX + y||Q = (/S2 [As, + s2|a<ir)1/<\ we have, by
Property 1.2.17,
e|ax + y|p = 11 ax + y||££|y0|p. (2.7.10)
Now
4rE\\X + Y\p = PE{XX + Y)<p~1>X
dX
and
\ (p/a)-l
n f r \uv«;-' r
= |(/ |As,+s2|°drJ a (\s\+S2)<a-x>sidr.

2.8
COVARIATION NORM
95
Use (2.7.10) and set A = 0 to obtain
£Xy<P-i> = f f \sj\ajp\ f f Sls<a-l>dr\E\Yo\P
= \\Yra-a{X,Y)aQE\Y0\r
= \\Y\\ZaiX,YFaE\Y\r,
which proves the result. I
2.8 Covariation norm
We know that any finite linear combination of SaS random variables is SaS.
Let <SQ be a linear space of jointly SaS random variables. When a > 1, the
covariation induces a norm on Sa.
Definition 2.8.1 The covariation norm of X S Sa, a> 1, is
\\x\\a = {[x,x\aya.
Corollary 2.7.6 implies that the covariation norm is equal to the scale parameter:
Property 2.8.2 IfX ~ Sa{a, 0,0) with a > 1, then
\\X\\a = a.
Proposition 2.8.3 || • ||Q is a norm on Sa. Convergence in \\ ■ \\a is equivalent to
convergence in probability and convergence in IP for any p < a.
PROOF: (a) Let X ~ Sa(ax, 0,0). To see that || • ||Q is a norm, note that
1. ||X||a = 0 if and only if X = 0 a.s. (by Property 2.8.2).
2. aX - SQ(|a|ax,0,0),sothat j|aX||a = |aiax = |o| ||X||Q.
3. If X\ and X2 are jointly SaS with spectral measure T, then
|jXl+X2||a = (TXt+Xi
Sl + S2rr(ds)
- a
< (J \Sl\aT{ds)\/a + ^Js\s2\aT(ds)
ex, + aXi
l!X,|U + ||x2|U.

96 MULTIVARIATE STABLE DISTRIBUTIONS 2.8
(b) To prove that convergence in the covariance norm is equivalent to
convergence in probability, let Xi,X2l... € Sa, Y e Sa. By Property 2.8.2,
lim \\Xn - Y\\a = 0 is equivalent to lim axn~Y = 0, where ax -y is the
n—>oo n—too ™
scale parameter of Xn - V. Fix e > 0. Then, if Z ~ 5Q(1,0,0) (that is, Z is
standard symmetric a-stable), we have, by scaling,
P(\Xn - Y\ > c) = P(a*„_y|Z| > c) = f(|Z| > —1—) - 0
as n —> oo if and only if
lim aXn-Y = 0.
71—1>00
(c) To show that this is equivalent to convergence in Lp for p < a, note that
E\Xn - Y|* = £(ax„-y|Z|)p = o^B_yE|Z|" - 0
asn —> coif andonly if limn-.coCTXn-y = 0- B
Moreover, we have
Property 2.8.4 Let (X, Y) ~ SaS with 1 < a < 2. Then
\[X,Y]*\<\\XUY\\%-1.
Proof: If T is the spectral measure of (X, Y) then, by the Holder inequality
\[x,Y\a\ = i f 8ls<a-i>nds)\
J s2
< (J |5,rr(dB))l/0 (Js |S2|(tt-1)(1-I/a)"r(ds)
= ,x^(1-1/o)
= \\x\\a\\Y\\rl- ■
Remark. Whereas the covariation norm is not defined when a < 1, the scale
parameter a exists for all possible values of a. The proof of Proposition 2.8.3
allows us to conclude that for any 0 < a < 2, aXn-Y -* 0 is equivalent to
convergence of Xn to Y in probability and in IP for p < a.
This result, in fact, is a natural one. For given a, the distribution of X ~
Sa(ex.0,0) depends only on the scale parameter ax- Since aax = |a|<7x> one
has
E\X\" = ax£|X0|p, (2-8.1)
1 — I/CK

2.9
JAMES ORTHOGONALITY
97
where Xq ~ Sa{ 1,0,0), so that convergence of ax to zero is equivalent to
convergence in probability or in Lp.
It is sometimes convenient to use the notation
\\X\\a = ox, 0<a<2,
to denote the scale parameter of a SaS random variable X.
2.9 James orthogonality
Property 2.7.11 states that if two jointly SaS random variables X and Y are
independent, then their covariation [X, Y]a is zero. The converse holds in the
Gaussian case q = 2 since [X, Y}2 = \Cov{X, Y) = 0 implies that X and Y
are independent, but in the non-Gaussian stable case a < 2, [X, Y]a = 0 does
not imply independence (see Example 2.7.12). What, then, is the meaning of
[X,Y]Q = 0whenl < a < 2?
Suppose, firstly, that a = 2. Jointly Gaussian mean zero random variables X
and Y can be viewed as vectors in the space Lq(Q, !F, P) of all random variables
with zero mean and finite second moment. The space Lq(Q, T, P) is a Hilbert
space with scalar product
{X,Y) := Cov(X,Y) = EXY.
The Gaussian random variables X and V are independent if and only if they are
orthogonal in Lg(Q, T, P).
Now let 1 < a < 2 and consider Sa, a linear space of SaS random variables.
We noted in the preceding section that Sa is a normed space, endowed with the
covariation norm || • ||a. One has \\X\\% = [X, X]a, but the covariation [X, Y]a
does not generally define a scalar product in Sa. The space Sa is, therefore, not a
pre-Hilbert space (i.e., a linear (vector) space endowed wkh a scalar product).
The classical notion of orthogonality is valid only in (pre-)Hilbert spaces.
Several alternative definitions of "orthogonality" can be introduced in normed
vector spaces. One of them is James orthogonality (James (1947)).
Definition 2.9.1 Let E be a normed vector space. A vector x € E is said to be
James orthogonal to a vector y 6 E (i -L j y) if for any real A
||x + Ay||>||x||.
In general, x ±.j y does not imply y Lj x. The relation ||ir + Ay|| > ||x|| for
all real A means that the vectors x + Ay, A € R, belong to a supporting hyperplane
of the sphere {s : ||s|| = ||a:||} at the point x.

98
MULTIVARIATE STABLE DISTRIBUTIONS
2.9
To see that James orthogonality is a natural extension of the usual notion
of orthogonality, suppose that the space E is pre-Hilbert and hence has a scalar
product. Let x,y € E and write x JL y if x is orthogonal to y in the usual sense.
Then
x±y =s> ||2 + Aj/||2 = ||x||2 + ||A2/||2 > ||a:||2 for any A
=*• z ±j y.
Observe, moreover, that if x is not orthogonal to y, then ||a; + Ay || can be greater
or smaller than ||x||, depending on the value of A. Hence,
x ± y <=> x ±jy
when E is (pre-)Hilbert.
If E is a normed vector space which is not pre-Hilbert, x ± y is not defined,
but x -L j y is well defined because the definition of James orthogonality involves
relations between norms and not between scalar products.
Now, let E be <SQ, 1 < a < 2, endowed with covariation norm || • ||Q.
Proposition 2.9.2 Let X and Y be jointly SaS random variables with a > 1.
Then
[X,Y]a=Q
if and only ifY is James orthogonal to X (symbolically Y ±j X), i.e.,
\\XX + Y\\a>\\Y\\a
for every real A.
PROOF: Suppose [X, Y]a = 0 and let A be arbitrary. Then, by Property 2.8.4,
\[\X + Y,Y}a\<\\XX + Y\\a\\Y\\Tl.
On the other hand, using Property 2.7.7, we have
[AX + Y,Y)a = \[X,Y]a + [Y,Y]a =-\\Y\\%,
since [X,y]Q = 0. (|| • ||° denotes the ath power of the covariation norm.)
Therefore,
\\Y\\Z<\\\X + Y\\Q\\Y\\r\
i.e.,
||AX + Y||Q>||Y||a.

2.9
JAMES ORTHOGONALITY
99
To prove the converse, let Y be James orthogonal to X and let T be the spectral
measure of the SaS vector (X, Y). Suppose that [X, Y]a < 0. Then
/ sis^a-1>r(ds)<0. (2.9.1)
For real A, define
g(X) = \\XX + Y\\Z = / |As, + s2\aT(dS).
Js2
We have, clearly,
g'(X) = a f Si(XSl + s2)<a-l>T(ds).
Js2
The fact that a > 1 implies that g' (A) is a continuous function of A. Since (2.9.1)
can be restated in the form g'(0) < 0, we conclude that g'(X) < 0 in some interval
(0, a). Therefore, for any 0 < A < a, we have
\\XX + Y\Fa = g(X)<g(0) = \\Y\\°,
contradicting the James orthogonality of V to X.
The case [X, Y)a > 0 is similar. I
The following proposition characterizes the SaS random variables for which
James orthogonality is a symmetric property.
Proposition 2.9.3 Let 1 < a < 2 and let Sa be a linear space of jointly SaS
random variables with dimSa > 3. Then the following four statements are
equivalent:
(i) Sa has the property:
X,Y£SQ, [X,Y]a=0 => [Y,X]a=0.
(ii) Sa has the property:
X..Y,ZeSa, [X,Y]a = Q, [X,Z]a=0 =» [X,Y + Z]a=0.
(Hi) There is an inner product (.,.) onSa such that
||X||Q = (X,X)'/2, X$Sa. (2.9.2)
(j'v) Sa consists of jointly sub-Gaussian SaS random variables.

100
MULTIVARIATE STABLE DISTRIBUTIONS
2.9
PROOF: Using Proposition 2.9.2, we can rephrase statements (i) and (ii) as
follows:
(i) James orthogonality in Sa is symmetric.
(ii) James orthogonality in Sa is additive on the right.
Since dim<SQ > 3, we conclude using Theorems 1 and 2 of James (1947), that (i),
(ii) and (iii) are equivalent. We shall now prove that (iii) implies (iv).
Set Rx,y - (X,Y), X,Y € SQ, and let {Gx, X € Sa} be a zero mean
Gaussian process having Rx,y as a covariance function.
Let A be a positive a/2-random variable, independent of the process
{Gx, X e Sa}, having Laplace transform
Eexp(--yA) = exp{-7a/2}. 7 > 0.
Define
We want to show that
{Zx, XeSa} = {X,XeSa}. (2.9.3)
Let X\, X2, ■■■-, Xn £ <SQ, and let bx, 62,..., 6„ be real numbers. Note that
n
Var(G^,6,x,-E^G^)
i=i
= (2>*J. £>J*i) -2J2{Y,Wi> bkXk)+Y,J2<!>jXj,bkXk)
j=l j-l fc=l j=l j=l fc=l
= 2 E £ bMx„ xk)~2j2J2 hMXj, xk) = 0.
j=\ fc=l j=l fc=l
Therefore, Gs^x* = E"=i 6jgx, as- so that
%-,^=i>^a-s- (2-9-4)
foranyXi,...,Xn £ <Sa, and any real 6,,. ..,6n.
To prove (2.9.3), it is enough to show that the distribution of any linear
combination of the random variables on the left-hand side of (2.9.3) coincides with
the distribution of the corresponding linear combination of the random variables
on the right-hand side of (2.9.3). Because of (2.9.4) we need only to show that

2.9
JAMES ORTHOGONALITY
101
Zx — X for any X € Sa. Since both X and Zx are SaS random variables, we
have only to show that
\\x\\a = \\zx\\Q,xesa.
On one hand,
by (2.9.2). To compute ||Zx||Q> recall that if G is zero mean normal, then
M1/2G||2 = 2-Q/2(Var G)a'2 (see Example 2.7.4). Hence,
||Zjt IL = \\V2Al/2Gx\\« = (VarGx)1/2 = Rfx,
proving that \\X\\a = ||ZxlU- Therefore, Sa consists of jointly sub-Gaussian
SaS random variables.
Finally, if (iv) holds, then (i) follows directly from (2.7.4). This completes the
proof. I
Remark. When dimSQ = 2, then Proposition 2.9.3 fails. More precisely, (ii) is
in general not equivalent to either (i), (iii) or (iv). For example, let
<Sa = {all + bV : a,6 real},
where U and V are i.i.d. SaS with 0 < a < 2. Let X = a\U + b\V and
Y = azU + &2V with non-zero coefficients ait bi, a.2, bi- Then
[X,Y]a = [o,C/ + 6,V,a2t/ + 62V]a
implies
Since a ^ 1 in general,
b2 /a,\<3tT> A 61 fa2\<^>
& [Y,X]a=Q
and hence (i) fails. Therefore, (iii) and (iv) fail as well because they both imply (i).
On the other hand, by (2.9.5), for a fixed X € <Sa and hence for a fixed a\ and b\,
the collection of all random variables W belonging to Sa such that [X, W]a = 0
is a (one-dimensional) linear subspace C of Sa. Therefore,
[X,Y]c = 0 and [X, Z]a = 0 =► Y 6 C, Z € £

102 MULTIVARIATE STABLE DISTRIBUTIONS 2.9
=> Y + ZeC =» [X,Y + Z]a = 0
and hence (ii) holds.
Note that one has (iii) <=> (iv) even when dim Sa = 2, because the argument
establishing this equivalence in Proposition 2.9.3 does not use dimensionality.
If dim <SQ = 2, then either (iii) or (iv) imply (i), but we do not know whether
(i) is equivalent to (iii) or to (iv).
The following proposition characterizes the spaces <SQ of SaS random
variables in which the normalized covariation is symmetric.
Proposition 2.9.4 Let 1 < a < 2 and let Sa be a linear space of SaS random
variables. Then the following two statements are equivalent:
(i) Sa has the property
X,YeSa,\\X\\a = \\Y\\a =* [X,Y]a = [Y,X]a.
(ii) Sa consists of jointly sub-Gaussian random SaS variables.
Proof: Let
(X,Y) = \\Y\\2-a[X,Y}a.
It is easy to check that this is an inner product on <SQ if (i) holds. We can then
apply the same argument as in the proof of Proposition 2.9.3. Finally, by using
Example 2.7.4, we conclude that (ii) implies (i). I
The next result due to Cambanis, Hardin and Weron (1988) is an immediate
consequence of Proposition 2.9.3 if dim<SQ > 3. It holds also, however, if
dim <Sa = 2.
Proposition 2.9.5 Let 1 < a < 2, and let Sa be a linear space of SaS random
variables with dim<SQ > 2. Then the following two statements are equivalent:
(i) Sa has the property
X,Y eSa, [X,Y}a =0 =* X,Y are independent.
(ii) a = 2, i.e., Sa consists of zero mean Gaussian random variables.
Proof: Suppose that (i) holds and choose an arbitrary non-zero X S Sa. Since
dim«SQ > 1 we may find a Z e Sa such that Z =£ XX for any real A. Let
0 = [Z, X]a/\\X\\%. Then [Z - (3X, X]Q = 0 so that, by assumption, Z - @X
is independent of X. Since both Z - /3X and X are SaS and non-zero, they can
differ only by the value of their scale parameter. There is, therefore, a non-zero

2.10
CODIFFERENCE
103
constant b such that Y = b(Z — (3X) has the same distribution as X, so that X
and Y are i.i.d. random variables. Note that by Properties 2.7.7 and 2.7.15,
[X + Y,X-Y\a
= [X, X\a + [Y, X}a - [X, Y}a - [Y, Y]a
= ira-ire=o-
Therefore X + Y and X — Y are also independent. But X ~ Sa (c, 0,0) for some
a > 0. Therefore, by the independence of X + Y and X - Y and of X and Y,
exp{-2Q<7Q} = £expi(2X)
= £expi[(X + Y) + (X-Y)]
= £expi(X + y)Eexpi(X-Y)
= (EexpiX)4
= exp{-4aa},
which implies 2a = 4 and a = 2. I
2.10 Codifference
The covariation was defined in Section 2.7. Here, we introduce a different
measure of bivariate dependence called the codifference which, like the covariation,
reduces to the covariance when a = 2. Whereas the covariation may not be
defined for a < 1, the codifference is defined for all 0 < a < 2.
Definition 2.10.1 The codifference of two jointly SaS, 0 < a < 2, random
variables X and Y equals
tx.y = \\X\\Z + \\Y\\Z-\\X -Y\\%, (2.10.1)
where ||X||a, \\Y\\a and ||X - Y\\a denote, respectively, the scale parameters of
X,YandX-y.
We shall be mainly interested in investigating the asymptotic behavior of
r{t) - TX(t),x(0) as t -> oo, where {X{t), t € R} is a stationary SaS stochastic
process. In this section, however, we consider some elementary properties of
Tx,y. where (X,Y) is a given SaS random vector. Observe firstly that, in
contrast to the covariation, we have:
Property 2.10.2
TX,Y = TY,X •
Like the covariation, the codifference reduces to the covariance when a = 2:
Property 2.10.3 If a = 2, then tx,y = Cov (X, Y).

104 MULTIVARIATE STABLE DISTRIBUTIONS 2.10
PROOF: When a = 2, \\X\\l = ^Var X and hence tx,y = |{Var X + Var Y -
Vai{X-Y)} = Cov{X,Y). I
The codifference, like the covariation, vanishes when the random variables
are independent:
Property 2.10.4 // X and Y are independent, then tx,y = 0. Conversely, if
tx,y — 0 and 0 < a < 1, then X and Y are independent.
PROOF: If X and Y are independent, then s\S2 = 0, Tx,y a.e. (Example 2.3.5),
where Fx,y is the spectral measure of (X, Y). Hence,
II*-112
= / |Sl-S2|QrX,y(ds)
= / |ai|arx,v(ds)+ / \s2\arx,Y(ds) (S,s2 = 0, rx,y a.e.)
JS2 JSi
= imi2 +mis,
i-e., tx,y = 0.
Further, if 0 < a < l.then \s\— s2\a < |si|Q + |s2|Q, with equality only when
s{s2 =0. The preceding computation shows that if tx,y = 0, then S]S2 = 0
Tx.y-a.e., and so X and Y are independent. I
When 1 < a < 2, t\,y = 0 does not imply that X and Y are independent
(Exercise 2.27).
Property 2.10.5
TX,Y < \\X\\aa + IIYIIS
and
JO 1/ 0<Q< 1,
TX'y - \(1 - 2Q-')(||X||S + MS) if 1 < a < 2-
Proof: The upper bound follows from ||X - y||Q > 0.
The lower bound on tx,y depends on a. If 0 < a < 1, then || • ||g satisfies
||X - y US < H*IIS + Hull's implying rx,Y > 0. If 1 < a < 2, || ■ ||a is a norm,
so ||X - yUS < (||X|U + l|yID* < 2Q-' (||X|IS + l|y IIS) by (2.7.6), implying
rx,y>(l-2a-1)(ll*IIS + l|y|IS)- ■
The upper bound is achieved when X = Y. The lower bound is achieved for
X and Y independent if 0 < a < 1, and for X = -Y if 1 < a < 2. If X and Y

2.10
CODIFFERENCE
105
are both standardized (\\X\\a = \\Y\\a = 1), then
0<rx,y<2 if 0<a<l,
2(1 - 2Q-') < tx,y < 2 if 1 < a < 2.
When a = 2, this is exactly equivalent to -1 < Corr {X,Y) < 1, where Con-
denotes the correlation coefficient.
The next property shows the significance of the codifference.
Property 2.10.6 Let {X,Y) and {X1 ,Y') be two SaS, 0 < a < 2, random
vectors with \\X\\a = \\Y\\a = ||X'|U = ll^'lla- Then
TX,Y < TX',Y'
implies that for every c > 0
P{\X -Y\>c}> P{\X' - Y'\ > c}.
Conversely, if the last relation holds for some c > 0, then tx,y < "Of'.y.
PROOF: Suppose that ||X - Y\\a ^ 0 and \\X' - Y'\\a ± 0 (the property holds
trivially otherwise). The inequality tx.y < tx',y' is equivalent to j|X — Y\\a >
\\X' — Y'\\a. This in turn is equivalent to
P{\X-Y\>c} = P{\\X-YV\X-Y\>\\X-Y\\~lc}
= P{\\X'-Y'\\Zl\X'-Y'\>\\X-Y\\Zxc}
= P{\X' -Y'\> \\X' -Y'\\a\\X -Y\\-lc}
> P{\X'-Y'\>c}. ■
The inequality P{\X - Y\ > c} > P{\X' - Y'\ > c) has the following
interpretation: the random variables X' and Y' are less likely to differ than X and
Y and so are "more dependent." Thus, the larger t, the "greater" the dependence.
The codifference shares an additional property with the covariance.
Property 2.10.7 Let (X\,..., Xj) be a SaS random vector. Then the matrix
{jXj.Xj i h 3 — 11 • • ■ > d} is non-negative definite.
Proof: We must show that for any integer d and any u\,...,Ud € R,
Si=i Sj=i TXi,Xj uiui ^ °- Let r ^ me spectral measure of the SaS random
vector (Xu...,Xd). Then

106 MULTIVARIATE STABLE DISTRIBUTIONS 2.10
d d
= / EX>*la + M° -1* -*ila)«i«jr(ds) > o
JSd i=1 j=i
because of the following lemma:
Lemma 2.10.8 For any 0 < a < 2 and n > 1, the function
{A(sus2) = |s,|Q + \s2\a - |s, -s2|a}, s,,S2 € Rn,
is non-negative definite.
PROOF: We must show that for any si,...,sm € Rn and real uu...,um,
J2iL\ X)JLi Msi> Sj)uiUj > 0. The proof is in three steps.
Step 1. View each u, as a mass at the point Sj, i — 1,..., m. Now add a mass
wo = - YliLi ui at tne origin so = 0. Then YaLo ^i = 0 and
m m mm
EE<Na + i^r - isi - sj-Dui^ = - EE is< - 8jia*ui
t=l j=I t=0 j=0
since, for example,
mm mm
EENQuiu;i = EiSiiauiEuJ
t=i j=\ »=i j=i
= -52|Si|Qttitlo
t=l
ElS* -So|aUiU0.
i=0
Step 2. For any constant c > 0,
i=0 j-0 i=0 j=0
m m
= -C^2^|Si-Sj|QUiUj+0(c),
i=0 j=0
as c tends to 0. It is then sufficient to prove that
i=0 j=0
Step 3. The function </>(0) = exp{-c|0|a}, 0 < a < 2, 0 = (0,,..., 0„)
is a characteristic function (of a sub-Gaussian SaS random vector (Proposition
2.5.5)). It is, therefore, a non-negative definite function of the variable 0, i.e.,

2.11
EXERCISES
107
2.11 Exercises
Exercise 2.1 Let Y — p\X\ + pzXz where p\ and p2 are non-zero real numbers
and Xi ~ SQ. (<7j, 0i,Hi), 2=1,2 with aj < c*2. Find C and 03, such that
P(|V| > A) ~ C\-°"
as A —► 00.
Exercise 2.2 Prove Corollary 2.1.3.
Hint: Compare with Definition 1.1.4 and Relation (1.1.3).
Exercise 2.3 Let {Y^ = (Y,(n),..., Yd(n)), n > 1} be a sequence of random
vectors in Rd. Prove that {Y^n^, n > 1} is tight if the sequence of components
{Yfc(n), n > 1} is tight for each k = 1,... ,d.
Definition. The sequence {Y^n), n > 1} is tight if for every e > 0 there is a
bounded rectangle A such that P{Y^n) e A} > 1 - e for all n.
Exercise 2.4 Using the results of Section 2.2, show that the function ^(fli,62) in
2.2.1 cannot be a characteristic function for 1 < a < 2.
Exercise2.5 Let Xi ~ SQ(ctj,/%,()), i — 1,2,...d, be independent random
variables. Characterize the spectral measure of the a-stable random vector
(Xi,X2,... ,Xd)-
Exercise 2.6 Prove Proposition 2.3.9 in the case a^l.
Exercise2.7 Let XUX2,X3 be i.i.d. 5,(1,1,0) and let Y, = X\ - X2 and
Y2 = X2 - X3. Let T be the random measure of Y = (Yi,Y2). What are the
points on which T is concentrated? Do these points have equal weight? What is
the shift parameter of the random variable 6\Y\ + 62Y21
Exercise 2.8 Let Xt and X2 be jointly SaS with spectral measure T\ Find the
spectral measure of the vector (aXi + bX2, cX\ + dX2).
Exercise 2.9 Let s(fc' = {s\k),..., s^) 6 Sd, k = 1,..., m, and suppose that
the a-stable random vector X = (X\,..., Xd) has a shift vector /x° and a spectral
measure T concentrated on the points s^, k = 1,..., m, and — s(fc), k =
1,...,m. Example 2.3.6 states that for suitably chosen triples {o->K,Pk,Hk), k =
l,...,m, and a real matrix A = {ajk), j = l,...,d, k = 1,... ,m, we have
(Xu...,Xd)^(j2a^Yk,...,YJ^kYk),
fc=i fc=i

108 MULTIVARIATE STABLE DISTRIBUTIONS 2.11
where Yk ~ 5Q(«Tfc,/3fe,Mfe), k = 1,... ,m are independent random variables.
Verify that the choice
W+ _ ly-
and
aJk = (W+ + Wk-y'asf\j = l,...,d, fc=l,...,m,
will work, where
wk+ = r({8(fc>}), wfc- = r({-s(fe)}), k = i, ... ,m.
How should the /ifc, fe = 1,..., m, be chosen?
Exercise 2.10 Find two symmetric (around 0) random variables X and Y, whose
sum X + Y is symmetric, but not around 0.
Hint: Let Zj, j = 1,2,3 be independent with an Si(1,1,0) distribution and set
X = Zi - Z2 and 7 = Zi - Z3 (Chen & Shepp 1983).
Exercise 2.11 Find another example of a strictly 1-stable random vector which is
not the translate of a symmetric 1-stable random vector.
Hint: Build the measure T in (2.3.1) concentrated on a few points of Sd- Reduce
(2.4.1) to a system of linear equations.
Exercise 2.12 Let X be an a-stable random vector in Rd with characteristic
function
d
Eexp{i(6>,X)} = exp|-(6>TI0)Q/2 + 1^0^°},
where T denotes transpose, Z is a positive-definite symmetric d x d matrix and
where pt° = (^°, /x§,..., n°d) ^ 0. Is its spectral measure T symmetric? Is X
strict? Is X symmetric? (Distinguish between the cases a ^ 1 and a = 1.) Is
X - fj,° strict? Is it symmetric?
Exercise 2.13 Consider the following two symmetric measures on S3:
= 3Q/2
V3 V5 v/3\ J V3 v/3 V3
6It'T'tJ+'1--3-- 3'
+ [6(1,0,0) + 6(-1,0,0)]
+ [6(0,1,0)+ 6(0,-1,0)]
+ [6(0,0,1)+ 6(0,0,-1)]

2.11
EXERCISES
109
and
Show that the two SaS distributions having, respectively, T\ and T2 as their
spectral measures have identical two-dimensional marginals.
Exercise 2.14 The characteristic function <p(6\, 92) and density function f{x\, x2)
of an isotropic Cauchy random vector X = (Xj, X2) are given in Example 2.5.6.
Derive the density / from the characteristic function cf> by using three different
methods:
(a) Use the inversion formula for characteristic functions.
(b) Use the fact that (XhX2) = A^2{GUG2) where G\ and G2 are i.i.d.
Gaussian random variables and where A1/2 is independent of (G\, G2) and
has a L6vy distribution S\/2(cr, 1,0) whose density is given in Chapter 1.
(c) Apply the formula for the density function of a multivariate Cauchy
distribution given in Example 2.5.3.
Exercise 2.15 Let 4>(9) and /(x) be, respectively, the characteristic function and
the density function of the multivariate Cauchy distribution, as given in Example
2.5.3. Derive /(x) starting from 4>(9).
Exercise 2.16 Prove Proposition 2.5.7.
Exercise 2.17 Let X = Xt + iX2 be a complex SaS random variable which is
isotropic, and let \X | be its modulus. Find the asymptotic behavior of P{|X| > A}
as A —► oo.
Exercise 2.18 Show that the complex random variable X = X\ + iX2 is not
isotropic if X\ and X-> are dependent mean zero Gaussian random variables.
Exercise 2.19 Suppose (X\, X2) is a SaS vector in K2 with uniform spectral
measure T such that T(S2) = 1. Use polar coordinates to evaluate the joint
characteristic function of (X\, X2).
Exercise 2.20 Prove Properties 2.7.7 and 2.7.8 by using Definition 2.7.1.

110 MULTIVARIATE STABLE DISTRIBUTIONS 2.11
Exercise 2.21 Let Sa be a linear space of SaS random variables with 0 < a < 1.
ForX € Sa, define
a(X) = scale parameter of X.
Show that a(X)a is a norm on Sa if a = 1, and a quasi-norm (i.e., the triangular
inequality holds) if 0 < a < 1.
Exercise 2.22 One can define the covariation for a = 1 as
[X,y],= / a, sign (S2)r(ds),
where T is the spectral measure of the jointly 515 stable random variables X and
Y. Using this definition, verify that all the results in Section 2.7 also hold for
a = 1, but that the alternative definition 2.7.3 is not valid.
Exercise 2.23 Define the covariation norm for a = 1 using the definition in the
previous exercise. Verify that all the results in Section 2.8 also hold for a = 1.
Exercise 2.24 Using the previous exercise show that the "Only if" part of
Proposition 2.9.2 holds for a = 1. Does the "if" part hold as well?
Exercise 2.25 Prove Proposition 2.9.5 when dim<SQ > 3 by using Proposition
2.9.3.
Exercise 2.26 Show that tx,y = 0 if X and Y are independent SaS, without
using the spectral measure.
Exercise 2.27 For a given 1 < a < 2 construct an example of a SaS random
vector (X, Y) such that tx,y = 0 but X and Y are not independent.
Hint: Choose Tx,y so that sis2 = 0 Tx,y-a.e.
Exercise 2.28 Let A(x, y) = \x\a + \y\a - \x - y\a, 0 < a < 2, x, y G R. Here
is another way to show that A is non-negative definite.
Step 1. For some c > 0,
/oo
{l-cos(es))\s\-{y+a)ds.
•oo
Step 2. The function
B(a,b) — cos(a-6) - cos a - cos b + 1, a, 6 € R,
is non-negative definite.
Hint: 2B{a, b) = g(a)g(b) + gja)g(b) where g(a) = 1 - eia.
Step 3. Show that A{x, y) = cj^ B(xs, ys)|s|-(1+Q>cte and conclude that
A is non-negative definite.

Chapter 3
Stable random processes and
stochastic integrals
An a-stable stochastic process is a random element whose finite-dimensional
distributions are a-stable. This definition is given in Section 3.1, and it is used to
introduce the notion of a-stable stochastic integrals, i.e., integrals of non-random
functions with respect to an a-stable random measure. It is convenient to view
these integrals as a-stable stochastic processes parameterized by their integrands.
In this way, we can define them by specifying their finite-dimensional distributions
(Section 3.2). In order to show they are also bona fide integrals, we proceed in three
steps: we introduce a-stable random measures (Section 3.3), construct integrals
of simple functions with respect to such measures and show that a stable integral
is a limit (in probability) of integrals of simple functions (Section 3.4.)
In Section 3.5, we develop some basic properties of stable integrals. We show,
for example, in marked contrast to the Gaussian case a = 2, that two a-stable
stochastic integrals with a < 2 are independent if and only if their integrands
have disjoint support. We also state the representation theorem, whose proof
can be found in Chapter 13. The representation theorem states that an a-stable
random vector can be represented as an a-stab-le stochastic integral. This is why
stochastic integrals are a convenient tool for studying a-stable random variables,
vectors and, as we will see, processes as well.
Several important examples of stochastic integrals are presented in Section
3.6. These include the SaS L6vy motion, which is the counterpart of Brownian
motion for a < 2; the (SaS) Omstein-Uhlenbeck process, which is the a-stable
extension of the Gaussian process with the same name; the reverse Ornstein-
Uhlenbeck process, which is different from the previous one when a < 2; the
well-balanced linear fractional stable motion and the log-fractional stable motion,

112
STABLE PROCESSES AND INTEGRALS
3.1
two self-similar (scaling) processes that will be discussed more thoroughly in
Chapter 7. Sub-Gaussian processes and substable process are additional examples.
They are introduced in Sections 3.7 and 3.8, respectively.
We mentioned earlier two definitions of stochastic integrals, one as a stochastic
process, the second as a limit of integrals of simple functions. We consider also
a third definition, in terms of a series representation. The series representation
of an a-stable random measure is developed in Section 3.9, and it is used in
Section 3.10 to obtain the series representation of the stochastic integral. This
representation sheds light into the structure of a-stable stochastic processes. We
use it, for example, in Section 3.11 to show that a broad class of SaS processes,
those that satisfy the so-called Condition S, are in fact conditionally centered
Gaussian processes. These processes always admit an integral representation.
There exists a fourth representation of stable stochastic integrals. Since a
stable measure can be expressed in terms of a Poisson measure, it is possible
to define stable stochastic integrals directly as integrals with respect to Poisson
measures. This was Paul Levy's original approach, and we present it in Section
3.12.
3.1 Stable stochastic processes
We start with the definition of a stable stochastic process {X(t),t € T}, where
T is an arbitrary set, e.g., R, Kn, or a set of functions. The finite-dimensional
distributions of {X(t), t € T} are the distributions of the vectors
(X(ti),X(t2),...,X(td)),tut2,...,td£T,d>l.
Definition 3.1.1 A stochastic process {X(t),t € T} is stable if all its finite-
dimensional distributions are stable. It is strictly stable or symmetric stable if all
its finite-dimensional distributions are, respectively, strictly stable or symmetric
stable.
If the finite-dimensional distributions are stable then, by consistency they must
all have the same index of stability a. We use the term a-stable when we wish to
specify the index of stability.
The following theorem is an immediate consequence of Theorem 2.1.2 and
Theorem 2.1.5.
Theorem 3.1.2 Let {X(t),t €T}bea stochastic process.
(a) {X(t),t € T} is strictly stable if and only if all linear combinations
d
Y,bkX(tk), d>l, tut2,...,td<=T, &,,&2)...,&<Jrea/ (3.1.1)
fc=i

3.2 DEFINITION OF STABLE INTEGRALS AS A STOCHASTIC PROCESS 113
are strictly stable.
(b) {X{t),t € T} is symmetric stable if and only if all linear combinations
(3.1.1) are symmetric stable.
(c)lfa > 1, then {X(t),t € T} is a-stable if and only if all linear
combinations (3.1.1) are a-stable.
Example 3.1.3 a-stable Levy motion. A stochastic process {X(t),t > 0} is
called (standard) a-stable Le>y motion if
(1)X(0) = 0a.s.
(2) X has independent increments.1
0)X{t)-X{s) ~ SQ((t-s)1/Q,/3,0)forany0 < s < t < oo and for some
0<a<2, -1 </3 < l.2
Observe that the process X has stationary increments. It is Brownian motion
when a — 2. The a-stable L6vy motions are SaS when /3 = 0 and they are
1/a-self-similar (unless a = l,/3 j= 0), i.e., for all c > 0, {X(ct),t > 0} and
{cl/aX(t), t > 0} have the same finite-dimensional distributions.
The role that a-stable Levy motion plays among stable processes is similar
to the role that Brownian motion plays among Gaussian processes.3 We will
encounter many more examples of stable processes in the sequel.
3.2 Definition of stable integrals as a stochastic
process
The "stable integral" of a non-random function / will be denoted 1(f). In this
section, we define the family of stable integrals {/(/), / 6 F} as a stochastic
process indexed by a set F of functions. The definition will ensure that the integral
/(/) is a stable random variable for each integrand /, and that /(■) is a linear
functional, i.e., J(ai/i + a2f2) - a\I(f\) + a2I{h) a.s. for any fu f2 6 F and
ai,a2 £ R-
In order to define the stochastic process {/(/),/ 6 F}, it is enough to specify
its finite-dimensional distributions, show that they are consistent and then appeal to
Kolmogorov's existence theorem (Billingsley (1986), Theorem 36.2) to conclude
that {/(/), / € F} is a well-defined stochastic process.
'This means that the random variables Xfo) - X(ti), X(t3) — X(ta), • • •, X(t„) - X(t„_i)
are independent for any t\ < ti < ... < tn.
2Some authors also impose a condition on the path structure. However, in this chapter, all processes
are defined only in terms of their finite-dimensional distributions.
3 Some authors whose research focuses on a-stable Levy motion call it "the stable process." Such a
terminology is confusing and should be avoided. When a = 2, one distinguishes between "Brownian
motion" and a "Gaussian process." Similarly, when a < 2, it is necessary to distinguish between
"a-stable L£vy motion" and an "a-stable process."

114 STABLE PROCESSES AND INTEGRALS 3.2
We start with the specification of the finite-dimensional distributions. Let
(E, £, m) be a measure space and let
/?:£-[-l,l]
be a measurable function. Choose
La(E,£,m) ifo^l,
(3.2.1)
J"(m,/3) ifa=l,
where
LQ{E,S,m) = {/ : / is measurable, f \f(x)\am(dx) < oo},
Je
F(m,(3) = {/:/€ L'(E,£,m) and / |/(z)/?(x) ln|/(x)| |m(di) < oo}.
Je
It is easy to check that F is a linear space. (See Exercise 3.3.) We may and will
assume without loss of generality that m is cr-finite, because f £ F implies that
the support of / is contained in a region of E where m is cr-finite (see Exercise
3.1).
Specification of the finite-dimensional distributions:
Given f\,..., fd € F, we define a probability measure P/,,...,/,, in Rd by its
characteristic function, as follows:
(i)Ifa^ 1:
0/i,-,/d(0i>--->0<*) =
- d d
eXp{- / \flejfi(X)\a{1 ~ W*) Si§n (Z^/jW) tSn ™)m(dx)}>
(3.2.2)
(ii)ifa= 1:
<Pf,,-,fd(Sl>---,6d) -
d 0 d d
We must prove that <pf ,/,, is, indeed, the characteristic function of a
probability measure in Rd; we will show that, in fact, it is the characteristic function of

3.2 DEFINITION OF STABLE INTEGRALS AS A STOCHASTIC PROCESS 115
an a-stable distribution. This is accomplished by making a change of variables
that will transform the integral over E to an integral over the unit sphere Sd in Md.
Note that here m is not necessarily the spectral measure of an a-stable law
because E is not necessarily the unit sphere Sd- In fact, the advantage of the
representation (3.2.2) is that the same measure m is used for all values of d and
all functions f\,..., /d in F.
For notational convenience, set 6 = (8y,...,9d), f = (/i, • • •, /d) and let
ua(8, f(x),(3{x)) denote
d d
^Ojfjirf^-iPix) sign (X>/j(*)) tan
j=\ j=i
ua\
if a t^ 1, and
d
IX^i/i^Kl +»-j8(i) sign ($>£(*)) lnl^/^x)!)
j=i j=i j=i
if a = 1. Set also
E+ = {a:€£:X^/jb)2>0}.
Suppose, firstly, that a ^ 1. Then
0/i,•••■/<»(0|>--- >^d)
= exp|-/ uQ(0,f(a;),/0(i))m(dx) V
= exp< — / uQ (0,g(i),)3(i))mi(di)j (3.2.3)
where
o (x) = ^ , j = 1,..., d,
3 (Et.A-W2)172
and
m,(dx) = Qr/fc(x)2) m(di).
fc=i
Note that mi is a finite measure on {E+,£) since fk e La(E,£,m) for fc =
1,..., d. Moreover, £?=1 ^(x)2 = 1, for all x e E+. Writing

116 STABLE PROCESSES AND INTEGRALS 3.2
4>j fd{6u-:,Bd) = expj-/" uQ(8,g(x),l)1+^x)mi(dx)
-jf «a(91-g(I))l)^Mmi((ii)}1
we can make the change of variables Sj = gj(x) in one integral and Sj = -gj(x)
in the other, to conclude that <Pfu...jd(Oi,..., 9d) equals
exp{_ ] \J2 6isl\a (l - * sign (^ej8j) tan ™)T(ds)},
where the finite measure T on Sd is given by
T(A)=[ l±|^mi(dx)+/ iz|Wmi(dl)i (3.2.4)
V'M) 2 Jg-'(-A) 2
where A is a Borel set in Sd and
<T' (A) ={i££+:(s,(i) ffd(x)) 6 A}.
This shows that (3.2.2) is the characteristic function of an a-stable law on Rd
when a ^ 1.
In the case a — 1, we have to pay special attention to the presence of the
logarithm. Since
*\±vM\-*\plls:^\Mp*tr-
Relation (3.2.3) becomes
f d
<£/„..,/d(0i.--->^)=exp{- / «a(e,g(a:),/3(s))m,(dz)+i£0i/*$}r
l Je+ j=i
where g = (ffi, • • •, 3d) and mi are defined as above and where
$ = -l-fE fiix)0(x)\n (^fkix)2J m(dx), j=l,...,d. (3.2.5)
Proceeding as in the case a ^ 1, we conclude that 4>jl,... jd {9\,..., 0<i) equals
«p{- / lE^|(l+i-sign(^^Sj)ln|X:^Sj|)r(ds)+iX:^}.

3.2 DEFINITION OF STABLE INTEGRALS AS A STOCHASTIC PROCESS 117
This shows that (3.2.2) is also the characteristic function of an a-stable law on Kd
in the case a = 1.
To verify the consistency of the probability measures P/,,...,/d, note that for
any permutation (tt(1), ..., n(d)) of (1,..., d), we have
^CD.-./'OflC^O' • • • > 0*(d)) - 4>I\ fd(0U ■ ■ ■ ,9d)
and for any n < d,
4>f /„(01,"-,0rO=4>/ /„,...,/,(fll,-.-,*n,0,...,0).
This proves consistency.
By Kolmogorov's existence theorem, there is a stochastic process which we
denote {1(f), / € F} whose finite-dimensional distributions are given by (3.2.2).
1(f) is called the a-stable integral of f. The measure m is called the control
measure and the function (3 is called the skewness intensity.
We now establish some elementary properties of the integral. We will say that
a function / is integrable if it belongs to the relevant index set F in (3.2.1).
Property 3.2.1 For any integrable fu f2,..., fd, the integrals I(f\), 1(h), ■■•,
I{fd) are jointly a-stable with joint characteristic function given by (3.2.2); they
are jointly SaS if the skewness intensity is zero.
Property 3.2.2 Let f be integrable. Then 1(f) ~ SQ(cr/,j3/,/i/) where
crf = [jE\f(x)\am(dx)j ,
JEf(x)«»P(x)m(dx)
Pf fE\f(x)\«m(dx) '
( 0 ifa^l,
y-f = {
I -lJEf(x){3(x)\n\f(x)\m(dx) ifa = l.
Property 3.2.1 follows from the definition of the process {1(f), f € F}. Property
3.2.2 is straightforward (substitute 92 = • • • = 0d = 0 in (3.2.2)). Note that when
a ■£ 1, /(/) has zero shift parameter and is, therefore, strictly stable, but it is
usually not strictly stable when a = 1. /(/) is symmetric a-stable when jS(-) = 0.
The next property states that the integral /(•) is a linear functional.
Property 3.2.3 // f\ and f2 are integrable, then
J(ai/i + a2f2) = aj(fi) + a2I(f2) a.s.
for any real numbers a\ and a2.

118 STABLE PROCESSES AND INTEGRALS 3.3
PROOF: We musl show that
/(a,/, + a2/2) - a,/(/,) - a2I(f2) = 0 a.s.
For any real 6 we have
£exp {i0[/(a,/, + a2f2) - o,/(/,) - a2/(/2)]}
= Eexp{t[flJ(a,/, +a2/2) - (a,0)I(/,) - (a20)J(/2)]}
= 1
by (3.2.2), because
0(ai/i + a2/2) - (a,0)/, - (a29)f2 = 0. I
3.3 a-stable random measures
It is most convenient to view a random measure as a stochastic process M(-)
indexed by sets A. The term "random measure" captures the two main
characteristics of M: the fact that {M{A\),...,M(Ad)) is a random vector and the fact
that M is additive, i.e., M(UiAj) = J2% M(Ai) a.s. if the sets Ai are disjoint.
This last relation does not imply that M is an ordinary signed measure for almost
every realization u>, since, for example, M(-, u) may not have bounded variation.4
As we will see in the next section, M can nevertheless serve as integrator in an
alternative definition /(/) = JE f(x)M(dx) of the stable integral.
We now turn to the definition of an a-stable random measure M. Denote
by {Q.,T,P) the underlying probability space and by L°(Q.) the set of all real
random variables defined on it. Let (E, £, m) be a measure space,
/3:E-[-l,l]
be a measurable function, and let
£o = {A € £ : m(A) < oo}
be the subset of £ that contains sets of finite m-measure.
Definition 3.3.1 An independently scattered <7-additive set function
M:£0-+ L°(Q)
4This is why some authors prefer to call M "random noise" instead of "random measure."

3.3
q-STABLE RANDOM MEASURES
119
such that for each A € £q,
1/q fA j3(x)m(dx)
M(A)~Sa[im{A))■/«, " — -,0
is called an a-stable random measure on {E,£) with control measure m and
skewness intensity (3.
Independently scattered means that if A\, A2, ■ ■ ■, Ak belong to £0 and are
disjoint, then the random variables M(yii),M(J42),... ,M(Ak) are independent.
a-additive means that if A\, A2.,... belong to £q, are disjoint and U^L, Aj € £0,
then
00 00
M(UAi) = EM^) a-s-
Note that we require the shift parameter fj, to be 0 for each M(A). This means
that the random measure M is characterized by a (non-random) measure m and a
function /3.
In order to show that the a-stable random measure M exists, we define
{M(A),Ae£Q}
as a stochastic process. Using the existence of a stochastic process {/(/), / S F)
established in Section 3.2 and setting M(A) = /(1a) for A e £0, yields the
existence of a stochastic process {M{A), A € £0} with the following finite-
dimensional distributions: for A\, A2,..., Ad e £0 and 9\, 62, ■ ■ ■, #<* real
numbers,
d
Y,0iM(Aj)~Sa{<r,P,ii), (3.3.1)
where
d
SEPW (Sj=1^UJ(x))<a>m(dx)
0 =
ifa#l,
/eIE^wwi"™^)
I -i/BE^.fliU^i^WlnlE^i^U/^Kdi) ifa = l.
Is such an M independently scattered? If the As in (3.3.1) are disjoint, then
fora# 1, by (3.3.1),

120
STABLE PROCESSES AND INTEGRALS
3.3
a
- d d
= exp{- / |^^lA.(x)|Q(l -i/3(x) sign (j^l^fc)) tan ^)m(dx)}
= exp{- £ l^l°m(^)(l - « JmU.} sign (9,) tan ^) }
d
= JJjSexpftfyM^)}.
j=i
This shows that M is independently scattered. The argument for a = 1 is similar.
Finite additivity follows from the linearity of the integral (Property 3.2.3) and the
equality XI,-=i ^A, — lul. a f°r disjoint A,s. But is M cr-additive? To prove
that it is, let AUA2,... d~£0, B = U£L, A,- € £0. We must establish that
oo
M(B) = Y,M(Aj) a.s.,
i.e..
or
lim YMiAj) = M(B) a.s. (3.3.2)
j=l
lim Y" M(AS) = M(B) in probability, (3.3.3)
r.—►no ^—J
n—>oo
because (3.3.2) is equivalent to (3.3.3), since the series 2"=i M(Aj) has
independent summands. By finite additivity, we have 5Z"_, M(Aj) = M(U"_, A,) a.s.
and, hence, almost surely,
n n oo
m(b)-Y,m(aj) = m(b)-m({Jaj)=m( (J ^),
j=l j=l j=n+l
by again using finite additivity. But M(WjLn+1Aj) ~ 5Q(crn,/?Tl)0), where
oo
<U/i)
oo
a" = m(
j=n+l
oo
= ^P MAO (cr-additivity of the measure m)
< co (because U£in+1 Aj C B € £o)-

3.4
CONSTRUCTIVE DEFINITION OF STABLE INTEGRALS
121
Since lim <jn = 0, we have M(U?L +lAj) p™ 0 as n —> oo and, hence,
71-^*00 •*
M(B)-Y4M(Aj)p^b0
as n —> oo, proving that M is cr-additive.
Definition 3.3.2 M is called a SaS random measure if the skewness intensity 0
is zero.
Example 3.3.3 Let M be an a-stable random measure on ([0, oo),S) with
Lebesgue control measure and constant skewness intensity fi(x) = /3, 0 < x <
oo. Let
X(t) = M([0,t]), 0<t<oo.
M([0, t]) is well defined because m([0, t]) < oo for any t > 0. It is easy to check
that the stochastic process {X(t), 0 < t < oo} is a-stable L6vy motion.
3.4 Constructive definition of stable integrals
The a-stable integrals 1(f) were defined in Section 3.2 as a stochastic process
indexed by the integrands /. In this section, we show that /(/) can also be
constructed as a bona fide integral, which we denote fE f(x)M(dx), where M is
some a-stable random measure. To define JE f(x)M(dx), we proceed as follows:
we approximate / by simple functions f(n\ i.e., functions that take only a finite
number of different values, we define JE f^(x)M(dx) and then take a limit in
probability as n -* oo. The limit is denoted jE f(x)M(dx). We finally show that
this definition coincides with the definition of the integral given in Section 3.2.
Let M be an a-stable random measure on {E, £) with a control measure m
and skewness intensity /3, and let S0 = {A £ £ : m(A) < oo}. Recall that we
want to define
US) = [ f{x)M{dx)
JE
for all measurable functions / : E -* 1R1 satisfying the condition
/ \m\°r
JE
*m(dx) < oo (3.4.1)
Je
and also the condition
/ \f(x)p(x) In |/(i)| m(dx) < oo (3.4.2)

122
STABLE PROCESSES AND INTEGRALS
3.4
if a — 1. We denote the collection of functions satisfying (3.4.1) and (3.4.2) by
F, as in (3.2.1). Observe that F is a linear space (Exercise 3.3).
As usual, for a simple function of the form f(x) = £"_, Cj 1 At (x) where A,
are disjoint sets belonging to £q, we define
1(f) = [ f(x)M(dx) ^Y^CjMiAj). (3.4.3)
Since the random measure M is independently scattered, the a-stable random
variables M(A\),...,M(An) are independent. Applying Properties 1.2.1 and
1.2.3, we conclude that /(/) ~ Sa(af,pf,Hf) with
<yf=(J \f(x)\am(dx))Ua, (3.4.4)
_ fBf(x)<a>P(x)mjdx)
f~ JE\f(x)\"m(dx) ' (3A5)
f 0 if a ^ 1,
N = < (3.4.6)
I -£/B/(*)/?(*) In |/(x)|m((k:) ifa=l.
The integral /(/) is obviously linear in the simple functions /.
Consider, now, a general / € F. We choose any sequence of simple functions
{/^}£Li possessing the following properties:
f{n) (x) -► fix) for almost every x € E, (3.4.7)
\f{n)(x)\ < B{x), for every n,x, and some 8 e F. (3.4.8)
Such a sequence always exists. We can take, for example,
' i ifi<f(x)<^,i = 0,l1...,n*-l,
/<">(*)=< -i if-^</(x)<-^i = 0,l,...,n2-l,
k0 if|/(i)|>n,
in which case 0 — \f\.
The sequence of the integrals J(/(n)), n = 1,2,..., is well defined by (3.4.3),
and we claim that it converges in probability. To show this, we must prove that
Hf^) ~ I(f^) converges to zero in the metric of convergence in probability
as n, m —> oo. But
J(/(n)) ~ J(/(m)) = Hf{n) - f{m)) ~ Sa(an,m,Pn,m,Hn,rn),

3.4
CONSTRUCTIVE DEFINITION OF STABLE INTEGRALS
123
where
1/q
' \ \f{n)(x) - fW(x)\am(dx)
— 1 < Pn,m < 1 and p,n,m equals 0 if a ^ 1 and
-- [ (/(n)(*) ~ f{m)(x))\n\f^(x) - f(m\x)\(3(x)m(dx)
*" Je
if a = 1. Tlierefore, convergence in probability of {I(fn),n — 1,2,...} will
hold if we prove that crn,m ->0and
Mn,m —► 0 as n, m —> oo. Obviously, (3.4.8)
implies that |/(7l)(x) ~ f^mHx)\ ^ 20(x), so that by the dominated convergence
theorem, <Jn,m —» 0 as n, m —> oo.
To show that pLn,m ~* 0 in the case a = 1, define
{ 0 ify = 0,
y\\ny\ ifO<y<e~\
e_1 if e-1 < y < r,
„ ylny if y > t,
where r is the solution of the equation t In t = e~'. Clearly, 1/; is a non-decreasing
continuous function and, moreover,
i>{y) = <
V'(y) > yI in i/l
for any y > 0. Note that for any 9 € F,
f j,{\6{x)\)\l3{x)\m(dx)
Je
(3.4.9)
/,
{|0(x)|6(e-',rl}
tf(|0(*)|)|/J(x)Mdx)
/
J{\(
+ I ^(\6(x)\)\P(x)\m(dx)
= e_l / |j3(x)|m(da;)
V{|9(x)|6(e-',T]}
< e m
/
i{x : \8(x)\ € (e-'.r]} + / \B(x){\n\6(x)\)0(x)\m(dx),
Je
+ I \e(x)(ln\6(x)\)p{x)\m(dx)
l{\0(x)me->,T}}

124
STABLE PROCESSES AND INTEGRALS
3.4
which is finite by (3.4.1) and (3.4.2). Now, the dominated convergence theorem
implies that
/ V(l/(n)(*) - /(m)(x)|)|/?(x)|m(dx) - 0 as n, m - oo,
Je
and by (3.4.9) this implies that /xn,m —> 0 as n, m —* oo. Therefore, the sequence
{/(/'")), n = 1,2,...} converges in probability, and we define
J(/)=pHmn^o0J(/(B)),
where plim denotes limit in probability.
This definition does not depend on the choice of the approximating sequence
/("). Indeed, if both /("> and g^ converge to / and satisfy (3.4.7) and (3.4.8),
then setting
( /<m) ifn = 2m,
h.™ = {
[ g(m) if n = 2m- 1
yields I{hW) p™b 1(f), which shows that the limits of I(f(m)) and I(g{m))
coincide.
Since convergence in probability implies convergence in distribution, we have
Proposition 3.4.1 Let aj,Pf,fif be defined, respectively, as in (3.4.4), (3.4.5)
and (3.4.6). Then
I(f)~Sa(af,f3f,fif),
i.e.,
(i)ifa^l:
Ezxp{i6I(f)}
= exp{- f |0/(x)|a(l -ip(x) sign (6f(x)) tan ^)m(dx)}; (3.4.10)
(ii)ifa=\:
£exp{z0/(/)}
= exp{- J \0f(x)\ (l + ilftx) sign (6f(x)) In |0/(s)|)m(dx)}.
We now show that 1(f) is linear in /. Let / € F, g 6 F and let
{/^}£Ln- {9^)^=1 te two sequences of simple functions satisfying (3.4.7)

3.4 CONSTRUCTIVE DEFINITION OF STABLE INTEGRALS 125
and (3.4.8) for / and g, respectively. Set h = af + bg and Mn> = afW + bg(n\
where a and b are two real numbers. Clearly, {h^}^^ is a sequence of simple
functions such that h^ (x) —> h(x) as n —> oo for almost every x, and
\h^\x)\ < |a||/(n)(z)l + l%(n)(*)l
< \a\ex{x) + \b\e2{x)
for some #i and B2 in F. Therefore,
7(/i) = plim^J^W)
= plimn_00(aJ(/W)+67(g(")))
= P^n-.oo^(fM) + plim^^WCs^))
= al(f)+bl(g),
proving linearity.
The linearity of the integral and Proposition 3.4.1 imply
Proposition 3.4.2 For any f\,...,fd in F, the characteristic function of the
random vector (i(/i), •.., /(/<*)) »•* given by (3.2.2).
This shows that the random vector (I(/i), ■.. ,I(fd)) is, in fact, a-stable and,
moreover, that the constructive definition of stable integrals is equivalent to that
given in Section 3.2. Therefore,
Proposition 3.4.3 For any f\,..., /<j in F, the random vector (/(/i),..., I(fd))
is a-stable with spectral measure T given by (3.2.4) and shift vector /u° equal to
zero ifa^X and given by (3.2.5) if a. = 1.
We adopt the terminology used in Section 3.2 and call /(/) = JE f(x)M(dx) the
integral of / with respect to an a-stable random measure M with control measure
m and skewness intensity /3.
Remark. If a = 1 and the control measure m is finite, then the space X(m, (3)
of measurable functions satisfying (3.4.1) and (3.4.2) actually becomes a Banach
space, as it is the intersection of the Banach space Ll (m) and the Banach (Orlicz)
space L\ogL(mp) of functions satisfying
f \f{x)\]n+\f{x)\mp(dx)<oo,
JE
where
In a if a > 1,
0 if 0 < a< 1,
ln+ a =

126 STABLE PROCESSES AND INTEGRALS 3.5
and where
mp(dx) = \/3(x)\m(dx).
We equip X(m, /3) with the maximum norm
ll/lli(m,/») = max (||/||L,(tn), H/ll
3.5 Properties of stable integrals
The following proposition relates the convergence of a sequence of a-stable
integrals to the convergence of the sequence of integrands.
Proposition 3.5.1 Let Xj = fBfj(x)M(dx), j = 1,2,..., and X =
JE f(x)M(dx), where M is an a-stable random measure with control measure
m and skewness intensity 0. Then
plim^^X,- = X
if and only if
lim / 1/jOr) - f{x)\am(dx) = 0 (3.5.1)
j'—00 Je
and in the case a = 1, if also
lim f (fjix) - f{x)) In \fj(x) - /(x)|)8(s)m(di) = 0. (3.5.2)
j-*00 Je
PROOF: The argument is similar to the one used in the constructive
definition of the integral. The convergence phra^^X,- = X is equivalent to
plimj-xxjCXj — X) = 0 and hence to convergence in distribution to zero of the
sequence {Xj - X, j — 1,2,...}. But the linearity of the integral and Proposition
3.4.1 imply Xj - X ~ Sa(aj,0j,(j.j), where
°j = (/ 1/i(2;) ~ ^l0™^) . J = 1.2, ■ ■ ■,
and
f 0, ifa^l
I ~\ JM*) - /(*))ln \fM - m\P(x)m(dx), if a = 1.
Therefore the convergence plim • ^X,- = X is equivalent to the convergence of
the sequences {aj, j = 1,2,...} and {fij,j = 1,2,...} to zero, establishing the
proposition. I

3.5
PROPERTIES OF STABLE INTEGRALS
127
Consider two a-stable integrals Xi = JE f\ (x)M(dx) and Xj =
JE h{x)M{dx). The next proposition expresses the covariation of X\ and X2 in
terms of/1 and /2.
Proposition 3.5.2 Let Xj = fEfj(x)M(dx), j = 1,2, where M is a SaS
random measure with 1 < a < 2 and control measure m. Then
[Xl,X2]a = f fl(x)f2(x)<a~x>m(dx). (3.5.3)
Je
Proof: Use the expression for the spectral measure T of (X\, X2) given in
Proposition 3.4.3. Applying Definition 2.7.1 of the covariation, we obtain
'Si
<a-l>
[x,,x2]Q= [ Sis<a-l>r(ds)
J Si
= JE+ (/?(*) + H{x)yn \{f2(X) + f2{x)y/2
= I h(x)f2(x)<a-l>m(dx)= [ fl(x)f2(x)<a-l>m(dx).
Je+ Je
MX) ( MX) ' (n(x) + f2(x))^m(dx)
IE-.
Remarks
1. Relation (3.5.3) can also be obtained as follows. Suppose that f\ and
f% are simple functions, i.e., /i(x) = 5Z?=i ui^Ai(x) and f2(x) =
2"=i vi^Ai{x)- By Property 2.7.7 (linearity in the first argument),
n n
[/(/.),/(/2)]a = Yl^[M(Ai),Y,VJM(Ai)}a
i=\ j=\
Since the M(Aj)s are independent, we also have additivity in the second
argument by Property 2.7.15. Hence,
[lUi),I{h)]a = J2J2uiv<*-l>{M(Ai),M(Aj)}a.
Independence yields [j¥(Ai),M(Aj)]Q = 0 if i # j, and
[M(Ai),M(Aj)]Q = ||M(Ai)||S = m(Ai) if i = j. Therefore
[/(/,), Hh)\a = jruiV?a-l>m(Ai)
t=i
Je

128 STABLE PROCESSES AND INTEGRALS 3.5
To derive this relation for all /, and f2 in LQ(m), approximate /, and f2 by
simple functions as in Section 3.4, use the Holder inequality to deduce that
/ \fi(x)\\f2{x)\a-lm(dx)
J E
'/a / r \ 1-1/a
< 00,
< (JE\fi(x)\am(dx)j U\f2(x)\am(dx)
and then apply the dominated convergence theorem.
For still another proof, see Exercise 3.4.
2. In the Gaussian case a = 2, we have [Xi, X2]2 = \ Cov(Xi, X2), so that,
when a = 2, (3.5.3) becomes
Cov(/(/,),I(f2)) = 2 / /,(x)/2(x)m(dx). (3.5.4)
Je
The independence of /(/,) = JEfl(x)M(dx) and I(f2) =
/e h(x)M(dx) is then equivalent to
/ /,(x)/2(x)m(dx)=0, (3.5.5)
a relation that can hold even when
m{support (/i) n support (fi)} > 0. (3.5.6)
(See Exercise 3.5.)
Independence of J(/i) and I(f2) in the case a < 2 imposes a much stronger
restriction on the functions /i and f2: they must have m-almost disjoint supports.
Indeed,
Theorem 3.5.3 Let Xl = fE fi(x)M{dx) and X2 = fB f2{x)M{dx) be two a-
stable integrals with respect to an a-stable random measure M with 0 < a < 2
and control measure m. Then X\ and X2 are independent if and only if
f\(x)f2(x) =0 m-a.e. onE.
Proof: For convenience, suppose a =fc 1. X\ and X2 are independent if and
only if for any real 0t and 92,
E&xv{i(8xXx+B2X2)} = £exp{i0iXi}£exp{i02*2}-

3.5
PROPERTIES OF STABLE INTEGRALS
129
But £exp{i(0iXi + 62X2)} equals
,2 2
exP\- / \J2ekfk(x)\a[\-iP(x)Sign(y29kfk(x))\m^]m(dx)},
Je fc=i fc=i
(3.5.7)
whereas i?exp{t#iXi}.Eexp{i02X2} equals
exp{- I |0,H/,(x)r [l - i/3(x) sign (fl,/,(s)) tan ™]m(dx)
- / l^n/2(x)r [l - *iS(x) sign (82fi(x)) tan ™]m(da:)p"
(3.5.8)
|f2|"|/2(a;)r I * ~ *PW sign (6>2Ma;)) tan — |m(dx) |
IE
Equating the moduli gives
[ \6lfl(x)+e2f2(x)\am(dx)
JE
= I^|Q / \fi(x)\am{dx) + \92\* [ \f2(x)\am(dx),
Je Je
which holds for all real 9\ and d2 only if
/,(x)/2(x)=0m-a.e. (3.5.9)
by Lemma 2.7.14. This proves that (3.5.9) is a necessary condition for the
independence of Xi and X2. It is also sufficient because if (3.5.9) holds, then the
imaginary parts of (3.5.7) and (3.5.8) are also equal. I
The preceding result is very useful and will often be used in the sequel.
Corollary 3.5.4 Let Xj' = JEfjM (dx), j — 1,..., d, be jointly a-stable. They
are independent if and only if they are pairwise independent, i.e., if and only if
fkt{x)fk2{x) — Om-a.e. on E for any subset {k\,k2} °/{l|2,. • • ,d}-
Proof: Independence clearly implies pairwise independence. To prove the
converse, use the following fact: X{,..., Xd are independent if and only if any
linear combination of {Xj, j 6 J} is independent of any linear combination of
{Xj, j € Jc}, where J is an arbitrary subset of {1,..., d}. Then apply Theorem
3.5.3 to conclude the proof.
More directly, pairwise independence implies by Theorem 3.5.3 that there are
disjoint Borel sets A\,...,A& such that
|{i:/i(s)^ 0)^1 = 0

130 STABLE PROCESSES AND INTEGRALS 3.5
for each j = 1,..., d. The joint characteristic function of X\,..., Xd factorizes.
For example, in the SaS case,
1 d r
Ecxp^djXj} = Eexp{iY,6j / fj(x)M(dx)\
r d
= exp{- / \)T6jfj{x)\am(dx)}
je j=1
d -
= exP{-E/ \0jfj(*)\a™(dx)}
d .
= exp{-£ / VjfjWmidx)}
d
= Y[Etxv{i9J / fj(x)M(dx)\
j=i ■'E
d
= J|£exp{i0jX,}.
i=\
This proves that X\,..., Xd are independent. I
The following proposition shows the effect of a change of variable.
Proposition 3.5.5 (Change of variable). Let Mm and Mv be either two a-stable
random measures with 0 < a < 2, a ^ 1 or two SIS random measures, with
identical skewness intensities and with control measures m and v, respectively,
satisfying
m(dx)
v{dx)
where r(x) > 0. Then
(r(x))Q, xeE,
f f(x)Mm(dx) = f f(x)r(x)Mu(dx) (3.5.10)
J e Je
foranyfeLa(E,m).
PROOF: Let f3(x), x € E denote the skewness intensity of the random measures
Mm and Mv. The result follows from the fact that the left-hand side and right-hand
side of (3.5.10) are both strictly a-stable with the same scale parameter
(J \f{x)\am{dx)^ ° = Q |/(x)r(x)|V(da:)N

3.5
PROPERTIES OF STABLE INTEGRALS
131
and same skewness parameter
JE(signf(x))\f(x)rp(x)m(dx)
fE\f(x)\"m(dx)
^ JE( sign f(x)r(x))\f(x)r(x)\°(3(x)v(dx) ^
SE \f(x)r(x)\"v(dx)
Remark. The conclusion of the proposition does not generally hold if a = 1 and
Mm is not symmetric stable because then the shift parameter of the left-hand side
of (3.5.10),
-- / /(aO/3(x)ln|/(x)Mdz),
71" JE
IE
does not necessarily equal the shift parameter
-- / f(x)r(x){\n\f(x)r(x)\)(3{x)v(dx)
K JE
of the right-hand side of (3.5.10).
We say that a process {X(t),t £ T} has the representation {Y(t),t € T]
if X = V, i.e., if the processes X and Y have the same finite-dimensional
distributions. We now turn to the representation theorem.
Recall that the distribution of an a-stable vector is characterized by its spectral
measure T on Sd and its shift vector /n° (Theorem 2.3.1). For a given a-stable
vector (X\,...,Xd) we want to find a measurable space {E,£), an a-stable
measure M denned on it, a sequence of measurable functions f\,..., fa, and a
constant vector tj in Rd, such that
{Xu...,Xd)±(j fl(x)M(dx),...,JEfd(x)M(dx))+71. (3.5.1
1)
The next theorem shows that this is always possible. Moreover, the space, the
random measure and the functions fj can be chosen in a special way.
Theorem 3.5.6 (Representation theorem in ]
Let X = (Xi, X2, ■ ■ ■, Xd) be an a-stable random vector in Rd.
(i) Let || -|| be an arbitrary norm o«Rd,5|j'" = {s: ||s|| = 1} the corresponding
unit sphere and let T\\.\\,ij&h be the corresponding spectral measure and shift
parameter as defined in Proposition 2.3.8. Then Relation (3.5.11) always holds
with:

132 STABLE PROCESSES AND INTEGRALS 3.5
{E,£) = {Sy, Borel o-algebra on Sy),
M has control measure m = r||.|| and skewness intensity /?(•) = 1,
fj : Si11-+R is defined by fj((sh...,sd)) = sj, j = l,...,d.
n = Mii.ii
so that
X^
[ 8iM(da), /" a2M(ds),..., f sdM(ds)\ +/*».„. (3.5.12)
(ii) //X is SaS, there are bounded measurable functions f\,fi,---,fd suc^
X = U' fi(x)M(dx), jf' /2(i)M(di), • • •, jf fd(x)M{dx) J ,
where M is an SaS random measure with control measure m(dx) = dx.
■(Hi) //X is strictly a-stable with a ^ 1, then there are bounded measurable
functions f\, /2,..., fd such that
X^ if fx(x)M{dx),j f2(x)M(dx),...,j fd{x)M(dx)) ,
where M is an a-stable random measure with control measure m(dx) = dx and
skewness intensity 0(x) = 1.
(iv) If a = 1, then there are bounded measurable functions f\, /2,..., fd
such that
X= (f /.(x)M(dx), /" Mx)M(dx),...,J fd(x)M(dx)\ +77,
w/iere M is a \-stable random measure with control measure m(dx) — dx and
skewness intensity /3(x) — 1, and where r) — (771,..., %) with
,...,d
ty = - /' /;(*)!"(£/*(x)2W + /.°, j = 1,2
Proof: (Part CO). In view of Proposition 2.3.8 it is sufficient to prove (3.5.12)
when || • || is the Euclidean norm on Rd. Then Sj", r||.|| and /z||| are the familiar
Sd, r and fi°. Use Proposition 3.4.3 to show that the joint characteristic function of

3.5
PROPERTIES OF STABLE INTEGRALS
133
the left-hand side and right-hand side of (3.5.12) coincide. (Observe in particular,
that when a = 1, the shift vector of the right-hand side equals
J Sd i t
'Sd fc=l
since£fc=i4 = 1.) ■
Parts (ii)-(iv) follow from a more general representation theorem, which will
be formulated and proved in Chapter 13. The idea behind their proof is to start
with the representation (3.5.12) on SJ|" and then make a change of variables in
such a way that S]J is mapped to (0,1) and the control measure of M becomes
Lebesgue.
Remarks
1. If || • || is the Euclidean norm, then Sd ', r||.|| and jujj,. are the usual Sd, T and
H°. Part (i) shows that one can always write down an integral representation
by using the pair (T, n°), which specifies the joint characteristic function.
2. A nice feature of the integral representation in parts (ii)-(iv) is that one
uses the same space E for all d > 1. The measure space, moreover, is
particularly simple since it equals ([0,1], B, dx).
3. Parts (ii)-(iv) effectively cover all cases of a-stable random vectors. In
the case where X\,X%,... ,Xd are not strictly a-stable and a ^ 1, the
representation (3.5.11) can be derived from Part (iii) of the theorem because,
by Property 1.2.6 and Corollary 2.4.2, there are constants 771,772,.. •, % such
that Xi - 771, X2 - 172, •.., Xd - T]d are jointly strictly a-stable.
4. Since the skewness parameter 0/ of JE f{x)M(dx) satisfies
JE\f{x)\a™-(dx) x6fi
• choosing /3(i) = C, -1 < C < 1 in parts (i), (iii) and (iv) of the theorem,
with C different from either 1 or -1 would unduly restrict the possible
values for /?/. With the choice /3{x) = 1, one has
JE(siznf(x))\f(x)\°m(dx)
Pf JE\f(x)\"m(dx)

134 STABLE PROCESSES AND INTEGRALS 3.5
5. The proof of the theorem in the Gaussian case a = 2 is straightforward.
If X\,X2,. ■■ ,Xd are jointly Gaussian with mean zero, then for some
constants a,jk, j = 1,..., d, k — 1,..., j,
X\ = anYj,
Xi — a,2\Y\ + 022^2,
X3 = 03iyi+a32l2 + a33l3,
Xd = ad\Y\ + a,d2Y2 + a^Ys H \-addYd,
where Yi,Y2,...,Yd are i.i.d. N (0,1). In this case the X,s are expressed
as a sum of i.i.d. N(0,1) random variables.
To express each Xj, j = 1,2,..., d, as an integral J0 fj(x)dM(x) where
M is a Gaussian random measure with control measure rri(dx) = dx,
choose
/^) = V2£Q'sl(^r-x<C)'
s=l x '
so that
Cov(/" fj{x)M{dx), f fk{x)M{dx))=2J fi(x)fk(x)dx
jAfc
s=l
6. The integral representation holds not only for a-stable random vectors in
Rd, but also for most a-stable stochastic process {X(t),t 6 T}, as we will
see in a Chapter 13. In that case, one may have to replace ([0,1], <B, da;) by
some general measure space (E,£,m).
1. Property 3.2.1 suggests that an a-stable random vector of the form
Xk= [ fk(x)M(dx),k=l,...,d,
Je
where M is a skewed a-stable measure, is, in general, not symmetric.
However, one can sometimes have X symmetric even with a non-symmetric
M. For example, part (iii) of Theorem 3.5.6 implies that any strictly a-
stable random vector with a ^ 1 (in particular, any SaS random vector

3.6
EXAMPLES
135
with a ^ 1) has an integral representation as above with a non-symmetric
M. A simple example (which includes a = 1) is
Xk= (l[fe_i>fc].(a:)-l[_fe,_fe+1](i))M(dx)) fc= l,...,d,
Jo
where M is an a-stable random measure with Lebesgue control measure
and skewness intensity /3 = 1. (To check that X is symmetric, verify that
all linear combinations of the components of X are SaS random variables.)
Pairwise independence of the components of a Gaussian random vector implies
independence of all the components. The same property holds for a-stable random
vectors.
Corollary 3.5.7 Let Xi,X2,---,Xd be jointly a-stable. If they are pairwise
independent, then they are independent.
Proof: By Theorem 3.5.6, we can assume without loss of generality that
Xk= fk{x)M{dx), k= l,...,d,
Je
where M is an a-stable random measure with control measure m and skewness
intensity /3 = 1. The result follows now from Corollary 3.5.4. I
3.6 Examples
Example 3.6.1 The SaS Levy motion.
Let
/•oo rt
X{t)= / \{x<t}M{dx)= / M(dx),t>0,
Jo Jo
where M is SaS on [0, oo) with control measure m(dx) = dx. Then
X(0) = 0 a.s.,
X{t) - X(s) = f M(dx) = M{[s,i\) ~ Sa{\t - s|1/Q,0,0).
IfO < t\ < t2 < ... < tn.then
(X{h) - X(tx),X(t3) - X(t2),. • • ,X(tn) - X(t„_i))
= IF M(dx), p Af(di),. • •, J " M(dx)\ .

136
STABLE PROCESSES AND INTEGRALS
3.6
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S0.5S cumulative
2000 4000 6000 8000 10000
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138
STABLE PROCESSES AND INTEGRALS
3.6
The components of this random vector are independent because the
integrands have disjoint supports. Hence, {X(t), t > 0} is a process that starts at 0,
has stationary independent increments and SaS finite-dimensional distributions.
Therefore, {X(t),t > 0} is the SaS L6vy motion (see Example 3.1.3). The
following figures display, for various values of a, 10000 realizations of i.i.d. SaS
random variables Cj (noise) together with their cumulative sums Xt = X^=i e»>
which approximate a SaS LeVy motion. Notice that the vertical axes do not have
the same scale.
Example 3.6.2 Moving averages
Let / be a measurable function on K1 satisfying J"f° \f(x)\adx < oo, 0 <
a < 2, and define
/oo
f(t-x)M(dx), teM1, (3.6.1)
-oo
where M is SaS with Lebesgue control measure (i.e. m(dx) — dx). A process
X(-) of the form (3.6.1) is called a SaS moving average process. X(-) is
stationary, because for any t\,..., td, h € R and real 9\,....,9d,
d -oo d
\\Y,6ix(ti+h)\\i = / \J2ejf^+h-x)\adx
Il>/fe-y)la*/
Example 3.6.3 Ornstein-Uhlenbeck process
Let A > 0 and M be a SaS random measure, 0 < a < 2 with Lebesgue
control measure. The process
X(t) = / e-x{t"x)M{dx), -oo < t < oo,
J — oo
is called an Ornstein-Uhlenbeck process. X(t) is a moving average process with
f{x) = e~All[0,oo)(z), -oo < x < oo,
and, hence, by Example 3.6.2 it is well defined and stationary. Note that for any
fixed s < t,
X{t) - e-^-^Xis) = J e-W-^Midx) (3.6.2)
f
J — (

3.6
EXAMPLES
139
is a SaS random variable independent of any linear combination
ELi *>**("*). uk < s, k= l,...,n. Therefore, X(t) - e-A('-*>X(s) is
independent of cr(X(u),u < s), the er-algebra generated by {X(u), u < s}.
This implies that the Ornstein-Uhlenbeck process is, in fact, a Markov process.
Recall that in the Gaussian case a = 2, the Ornstein-Uhlenbeck process is
the only continuous in probability stationary Markov Gaussian process (Breiman
(1968)). In the case 0 < q < 2 there is at least one more stationary Markov SaS
process. It is given in the following example.
Example 3.6.4 Reverse Ornstein-Uhlenbeck process
Let A > 0 and M be a SaS random measure, 0 < a < 2 with Lebesgue
control measure. The process
X{t) = / e-^-^Midx), -oo < t < oo,
is well defined and is called reverse (or fully anticipating) Ornstein-Uhlenbeck
process. Note that for any s > t
X{t) - e-A(-*>X(a) = [' e-^-^Midx). (3.6.3)
Using the same argument as in the previous example, we conclude that the reverse
Ornstein-Uhlenbeck process is also Markov.
We noted earlier that in the Gaussian case a — 2, all stationary Markov
processes are Ornstein-Uhlenbeck. This means that when a — 2, the Ornstein-
Uhlenbeck process and the reverse Ornstein-Uhlenbeck process are the same
process. (One can also verify this fact by computing the autocovariance function.)
It turns out that these two processes are different when 0 < a < 2. To see
this, let X\ and X2 be, respectively, the Ornstein-Uhlenbeck and the reverse
Ornstein-Uhlenbeck processes. Fixing s <t and using (3.6.2), we have
£exp{i(^,X1(s) + a2Xi(t))}
= Eexpji^, +02e-A(t-s))X1(S) + 02(X1(t)-e-^-s)X1(S))]}
= Eexp{*02(X,(t) - e-^-5)X((S))} • £exp{i(0, + 02e-^-3>)X,(S)}
= exp{-|02r j\e-^^)adx - |0, + 92e-^-^\a f (e-x^)adx]
= exp{-^ [A\92\a + B-a\B9{ + Be-x{t-s)92\a]},

140 STABLE PROCESSES AND INTEGRALS 3.6
where A = 1 - e-°A(t-s) and B = |1 + e-2*(*-*)|-i/2. Hence, the spectral
measure H of the SaS random vector (X\ {s),X\ (*)) is given by
H = ^[^(5{o,i} + ^{o,-i}) + B~a{6{BBe-Ht-,)) + 5{_B>_Be-M«-.)}) •
Similarly, the spectral measure T2 of the SaS random vector (X2(s), X2(t)) is
given by
r2 = ^^ [^(5{1,0} + 6{-l,0}) + B~a{6{Be->,(t-.):By + £{_Be-M«—>,_B})] ■
Since T\ ^ r2, the uniqueness of the spectral measure implies that
(Xi(s),Xi(i)) ^ (X2(s),X2(t)). Therefore the Ornstein-Uhlenbeck and the
reverse Omstein-Uhlenbeck processes are different for 0 < a < 2.
Example 3.6.5 Well-balanced linear fractional stable motion
Let M be SaS, 0 < a < 2, with Lebesgue control measure and consider
/oo
(|t - x\H~x'a - \x\"-l'a)M(dx), -oo < t < co, (3.6.4)
•oo
where 0 < H < 1, H ^ \/a. The process X is called the well-balanced
(symmetric) linear fractional stable motion. (We will encounter other linear
fractional stable motions in Chapter 7.)
The process X(t) has two important features. It is self-similar with
parameter H, i.e., for any c > 0, t\,...,td € K, (X(cti),... ,X(ctd)) =
(cHX(ti),..., cHX(td,)), and it has stationary increments, i.e., for any t€R,
{X{t) - X(0), -oo < t < oo} = {X{t + t) - X(r), -oo < t < oo}.
(See Exercise 3.9.)
In the Gaussian case a = 2, the process (3.6.4) is known as fractional Brow-
nian motion.
To verify that the process X(t) is well denned, we must prove that
/oo
\\t-x\H~^a-\x\H-^a\adx<oo.
-oo
This integral may diverge at (i) x — ±oo or at (ii) x = 0 or t. As x —» ±oo, the
integrand behaves like (\x\H~i/a~l)a = \x\Ha~1~a, which is integrable since
H < 1. As x -+ 0 or i, the integrand behaves like (|o:|H-1/0,)Q = la;!""-1,
which is integrable since H > 0. Hence the process (3.6.4) is well defined.
The process X(t) will be discussed more thoroughly later on, in the context
of self-similar processes.

3.6
EXAMPLES
141
Example 3.6.6 Log-fractional stable motion
Again suppose M to be SaS with Lebesgue control measure but assume that
1 < a < 2. The process
/CO
(ln|t-i|-ln|x|)M(da;), -co < t < oo, (3.6.5)
•CO
is called the (symmetric) log-fractional stable motion. It is self-similar with
H = \/a and has stationary increments (see Exercise 3.10). The value a = 2
is also admissible, but then (3.6.5) and (3.6.4) become identical processes (see
Exercise 3.11).
Recall that a-stable motion (Example 3.6.1) is also self-similar with H = 1/a
and has stationary increments (see Example 3.1.3). Log-fractional stable motion,
however, is a different process. To see this, note that the increments of a-stable
motion are independent but, by Theorem 3.5.3, the increments of log-fractional
stable motion are dependent since
/oo
(In I* + 1 — a:| — In |t — x|)Af (dx)
-oo
and the functions In \t + 1 + -| — In \t — -\ and In \t — -| — In | • | do not have disjoint
supports.
The log-fractional stable motion will be discussed more thoroughly in the
context of self-similar processes.
Example 3.6.7 Real stationary SaS harmonizable process
The finite-dimensional characteristic function of a real SaS harmonizable
process {H{t), t e T} is
Eexp{i(0iH(ti) + ■ • • + 0dH{td))}
f^fljcostjij 4- [Y^OjSmtjXj m{dx)\
■°° j=i j=i
= exp|-co/ 2^2_l^j^cos((t:j-tfc)x) m(dx)\, (3.6.6)
where m is a finite measure on (R, S) and co == ^ /0 (cos <j>)ad<j> is the constant
which appears in Corollary 2.6.5.5 The simplest way to represent the process
{H(t), t e T} is through the representation
/oo
eitxM{dx), t e T,
-OO
5The constant Cq can be incorporated in m.

142 STABLE PROCESSES AND INTEGRALS 3.7
where M is some complex-valued measure. This representation is defined in
Chapter 6.
3.7 Sub-Gaussian processes
Let {G(t),t € T} be a Gaussian process and let A ~ ^^((cos^)2/", 1,0)
where a < 2, so that Ee~^A = e"1'"1, 7 > 0 (see (2.5.2)). Assume that the
random variable A is independent of {G(t),t e T). The SaS process {X(t) =
A1/2G(t), t G T} is a sub-Gaussian process with an underlying Gaussian process
{<?(£),t S T}. Its finite dimensional projections, (Jf (ti),...,X(t<j)), d ^ 1,
are the sub-Gaussian SaS random vectors introduced in Section 2.5. What kind
of integral representation does the process {X(t) = A^2G(t), t S T} have?
Let (Q., T, P) denote the probability space on which {G(t), t € T} is defined.
We write {G{t),t € T} = {G(t,w), ( e T, u 6 £1} in order to make the
dependence on li explicit. However, to avoid possible confusion, we use in the
sequel the letter x instead of the letter u. Now, let M be a SaS random measure
on (£}, J7) with control measure P. The following proposition gives the integral
representation of X(t).
Proposition 3.7.1 The sub-Gaussian process {X(t) = A^2G(t), t £ T} has
the representation
{X(t),te T) L {jL= J G{t,x)M{dx), t e r}, (3.7.1)
where da = {E\Z\aY/a with Z ~ N(0,1).
Remarks
1. The right-hand side of (3.7.1) is somewhat unusual because E = O. and
m = P. Observe that although G(t,x) is not random, M is random, and
hence X(t) is random.
2. Theintegraliswelldefinedbecause/n|G(t,x)|QP(d:r) = E\G(t)\a < 00.
3. Any N(Q, o2) random variable G satisfies
a
a
E\G\a = E- V2)*'2 = daa{E\G\2)a'2.
4. One has (see Exercise 3.12),
(da)a = E\N(0, 1)|Q = 2a'2^-x,2T (^-) • (3-7-2)

3.8
SUB-STABLE PROCESSES
143
PROOF: Forany d> 1, tu...,td e T and 6>,,... ,6>d real,
Eex^iJ^Bj-S^ J G{thx)M{dx))
r 1 ^
= exp{~ _/n^a|^ J^djGitj^^Pidx)}
= exP{-<ir-E|^E^Gfe)r}
= exp{-(£|-L53^G^)|T/2}
j=i fc=i
d
by Proposition 2.5.2. I
i=i
(3.7.3)
3.8 Sub-stable processes
The Gaussian process G(t) is stable with index of stability a' = 2. What happens
if, in the preceding section, we replace G(t) by a process Y(t) that is stable with
a' < 2 and modify A accordingly?
Let {Y(t),t E T) be a Sa'S process with a' < 2. Fix a < a' and let
a' /a
*~MHg)) -1'0)
be independent of the process {Y{t),t € T}. Relations (1.2.3) and (1.2.4) ensure
that A has Laplace transform Ee~lA = exp{-7a/a'}, 7 > 0.
Proposition 3.8.1 The process
X{t)^Al/a'Y{t), t€T,
(3.8.1)
is SaS.

144
STABLE PROCESSES AND INTEGRALS
3.8
Proof: For any d > 1, tu...,td€T,b\,...,bd real, 9 real,
d d
' JSexpjiflJ^-Xfa)} = EE\exv{i6Al/a'f^bjYiti)}^]
j=i L j=i
= s«p{-4«i;w«i>i£}
= exP{-ierii5;&inti)C}. (3.8.2)
j=i
This is the characteristic function of a SaS random variable. (We use the notation
H^llcn a < 2 to denote the scale parameter of an a-stable random variable U.)
Since every finite linear combination of components of {X(t),t G T} is a
SaS random variable, the process {X(t), t 6 T} is SaS by Theorem 2.1.5 and
Definition 3.1.1. I
The SaS process X(t) is called a sub-stable process or sub-a-stable process
with underlying stable process {Y(t),t € T}. It is sub-Gaussian when a' = 2.
To find its integral representation, we proceed as in the sub-Gaussian case. Let
(Q, J7, P) denote the probability space on which {Y(t), t € T} is defined, let M
be a SaS random measure on (Q, T) with control measure P and let dPiQ< =
(E\Z\Py/? where Z ~ 5^(1,0,0).
Proposition 3.8.2 77k: sub-stable process X(t) = A^V^),* e T has the
representation
{X(t),t e T} ^ |d-'Q, | y(t,x)M(<ir), t e r}. (3.8.3)
PROOF: Proceeding as in the proof of Proposition 3.7.1, we have
d d
Sexpfc, / Y/6jY(tj,x)M(dx)} = exp{-d;°,B|^^(tj)!0}
= exP{-||£Wt;C}
i=i
by (2.8.1), which is equal, by (3.8.2), to £exp{i Y?j=\ &jX{tj)}. I
Remark. The process {X(t) = jnW(t,x)M{dx), t € T} is well defined for
any stochastic process {W(t),t 6 T} defined on (Q, .F, P) with E\W(t)\a < oo,
and for any a-stable random measure M on (Q, J7) with control measure P. The

3.9 SERIES REPRESENTATION FOR a-STABLE RANDOM MEASURES 145
random measure M does not even have to be symmetric. The resulting process
{X(t), t € T} will then be a-stable, but it is not in general in the form ApW(i)
where A is a random variable and p is a real constant.
3.9 Series representation for a-stable random
measures
The series representation described in Section 1.4 for one-dimensional actable
random variables can be extended to a-stable random measures, and it indicates
that a-stable random measures have essentially a "discrete" structure.
Let M be an a-stable random measure on {E, £) with finite control measure
m and skewness intensity /?(•). Denote by m the normalized measure
m
m =
m{E)'
Let {ri,r2,...} be a sequence of arrival times of a Poisson process with unit
arrival rate, and let {(Vl,7i), (^,72), • • •} be a sequence of i.i.d. random vectors,
independent of {T\, r2,...}, such that Vt has distribution man E and
P{li = l|v4) = 1 - P(li = -l|Vi) =
l+/?(Vi)
We have the following:
Theorem 3.9.1
{M(A),Az£}±
CO r - -1
{{Cam{E))l/a ]T nT-'^W e A) - b\a) I /3(x)fh(dx) +r,A,Ae £)
(3.9.1)
where Ca is given in (1.2.9),
' 0 if 0 < a < 1,
b^ = \ J",1//'- ° x~2 smxdx if a = 1,
and
' 0 i/a/l,

146 STABLE PROCESSES AND INTEGRALS 3.9
Proof: Fix A e £ and let w}A) = 7l-l(Vi e A), i = 1,2,... . Clearly,
w}A\i = 1,2
and satisfying
w}A\i~ 1,2..., is a sequence of i.i.d. random variables taking values -1,0,1
E\W{A)\a = P(Vi € A) = m(A),
EW[A) = f E(-ri\Vi=x)rh(dx)
J A
= J [P(7i = 1|V, = x) - P(7, = -1|V, = x)]m(dz)
= / (3(x)m(dx)
J A
and
£W,(>1) In |W,(A)| = Efi \{VX € A) In 1(V, € A) = 0.
We apply now Theorem 1.4.5 to conclude that the series
J2[r^7UanVieA)-b(r) j P(x)m(dx)\
converges a.s. to a Sa(aA, Pa, 0) random variable with
- _ ™(A)
aA - —F< '
and
_ E{WlA))<a> _ EW\A) _ JAP(x)m{dx) _ JAp(x)m(dx)
A ~ E\w\A)\<* ~ E\wlA)\a ~ m(A) ~ m(A)
Hence, by Property 1.2.3, the random variable
M(A) = {Cam{E))xlaY\liT-Ua\{Vi € A) - b\a) / 0{x)m(dx)
has a Sa((TA, Pa, 0) distribution with
aA = (Cam(E))<r% = m(A)
and
Pa^Pa
+ 7M
(3.9.2)
JAP(x)m(dx)
m{A)
To see that it has zero shift when a = 1, note that C\ = ~ and that tja
exactly compensates for the shift induced by the multiplicative constant C\m(E).
Therefore M{A) = M(A) for each A <E £.

3.9 SERIES REPRESENTATION FOR a-STABLE RANDOM MEASURES 147
We now prove that
(M(A,),..., M(Ad)) L (M(A,),..., M{Ad)) (3.9.3)
for disjoint sets A\,..., Ad. Let 6\,..., 9d be real numbers. By (3.9.2),
d
J'=l
(CQm(£))'/°£ liT7l/aY,e^Vi^Aj)-t)T/0j / 0(x)m{dx)
+E^-
j=l
Let iy< = 7iE?=i^lW 6 A,) = EjiiW^'' i = 1,2,... . Then the
sequence {Wj,i = 1,2,...} is a sequence of i.i.d. random variables satisfying
E\Wi\a = E>,rP(Vi e A,-)
i=i
d
= ^l^rm(^),
i=\
d
£(W,)<a> = £ f E((llej)<a>\Vl=x)fh(dx)
d r
= Efl/Q> / ^w^(^).
j=I -^
d /•
EW, = EflJ / /3(x)m(di),
and
£W, In |Wi | = jB ]T fljWr,(A,) In 53 #j^(
(Aj)
J=l
j=l
Applying Theorem 1.4.5 again, we conclude that the series
OO r d d „
£ -wrr^EffjlWe^-bj"^ / /3(x)m(dx)

148 STABLE PROCESSES AND INTEGRALS 3.9
has a .S'„ (ci, /?i, Mi) distribution, where
ca
^d
/?> =
Zl^0fa>JAil3(x)m{dx)
and
Mi
0 if a ^ 1,
- £;=1 fli In |flj | /A. 0(x)m{dx) if a = 1.
Hence, by Property 1.2.3, Ej=i fy-M(Aj) ~ SQ(o-2, /%, /i2), where
d
a? = (Cam(E))af = ^l^mCA,-),
0 if a 7M,
-1 £?=, *j 1" ft1 JAi P(x)m(dx) if a = 1,
and
M2
since Ci = -. Recalling that M is independently scattered and that each M(A)
is a-stable with scale parameter M(A)1/q, skewness m(A)~l fA P(x)m(dx) and
zero shift, we obtain by Property 1.2.1 that
d d
j=\ j=i
Since this is true for any real Q\,..., 92, we conclude that Relation_(3.9.3)holds
for disjoint sets A\,...,Ad. Because of the additivity of M and of M (for M, see
(3.9.2)), Relation (3.9.3) also holds for non-disjoint sets A\,...,Ad, concluding
the proof of the theorem. I

3.10
USING THE SERIES REPRESENTATION
149
3.10 A third definition of stable stochastic integrals
using the series representation
Let M be an a-stable random measure with 0 < a < 2. The series representation
for M given in Theorem 3.9.1 shows that the measure M is the sum of two
components: a random component consisting of signed random point masses
1iF~ placed at random points Vit and a non-random component proportional
to the signed measure ms(dx) = (3(x)m(dx). The non-random component is
absent when 0 < a < 1 and is equal to zero when M is SaS. Therefore,
if / is an integrable function, its integral /(/) = JE f(x)M(dx) should have
a representation as a series whose terms are the sum of two components: a
random term equal to 7,r~ f{Vi) and a non-random integral proportional to
JE f(x)ms(dx). The following theorem states that this is indeed the case. The
proof is similar to that of Theorem 3.9.1 and is left to the reader (see Exercise
3.14).
Theorem 3.10.1 Let {E,£,m) be a finite measure space and let M be an a-
stable random measure with 0 < q < 2, a finite control measure m and skewness
intensity f3. Let {T\, T2,...} be a sequence of arrival times of a Poisson process
with unit arrival rate, and let {(V\,~yi), (14,72), • •.} be a sequence of i.i.d. random
vectors such that V* has distribution fh = m/m(E) on E and
Pin = i|vi) = 1 - p(h = -i|vi) = i±ffl.
Assume also that the sequence {(Vi, 71), (1^,72), • • •} is independent of the
sequence {ri,r2,...}.
Then, for any function f S F (defined by (3.4.1) and (3.4.2)), we have
1(f) = S(f),
where S(f) is an a.s. convergent random series defined as follows:
(i)If 0<a< 1,
00
S(f) = {Cam(E))l'aYl*T7l,afW>
where Ca is given in (1.2.9).
(ii)lf a= I,
S(f) = lm(E) f;|W7(Vi) - b\l) J f(x)P(x)fh(dx)] + ij/,

150 STABLE PROCESSES AND INTEGRALS 3.10
where b\ ' is given in (3.9.1) and
r]f = -In (-m{E) J f f{x)/3(x)m(dx). (3.10.1)
(Hi) If 1 < a < 2,
°°
S(f) = (Cam(E)y/° £ [7iI7,A7W) - b^ I f(x)P{x)m(dx)},
i=i L ^ j
w/iere 6; ' is given in (3.9.1).
We refer to S(f) as the series representation of the stable stochastic integral 1(f).
It is clear that S(f) is linear in / € F. Therefore, for any /t,..., fd £ F and
any real numbers 9\,...,0<i,
Y,oksUk) = s X>A J = 7 E^A J = I>J(/0-
fe=l \fc=l / \fc=l / fc = l
We then immediately obtain the following:
Corollary3.10.2 ///,,/2,..,/,,eF,/Aen
(i(/1),Jr(/2),---,/(/d)) = (5(/,)I5(/2),...!5(/(i)).
The strength of this result allows us to regard S(f) as an additional definition of
stable stochastic integral.
Specifically, consider a finite measure space (E, £,m),a. measurable function
(3 : E —► [— 1,1] and a measurable function /:£-»R satisfying
/.
\f(x)\adx < oo,
E
and also, if a = 1,
[ \f(x)(ln\f(x)\)0(x)\m(dx)
JE
< CO.
We then define S(f) as in Theorem 3.10.1 and call it the a-stable integral of
f. The measure m is called the control measure and the function (3 is called the
skewness intensity of the integral. Corollary 3.10.2 implies that this definition of
stable stochastic integral is equivalent to the definitions given in Sections 3.2 and
3.4.

3.10
USING THE SERIES REPRESENTATION
151
Example 3.10.3 Here, we use the series representation of stable stochastic
integrals to gain insight into the structure of Levy a-stable motion.
Let {X(t),0 < t < 1} be standard Levy a-stable motion on [0,1], i.e., with
Z(l) ~ SQ{l,P,0). Example 3.1.3 shows that
{X(t),0<t< !} = {/ l[Q,t]{x)M(dx), 0<t< l},
where M is an a-stable random measure on ([0, l],B) with Lebesgue control
measure and skewness intensity @(x) = @. Therefore Corollary 3.10.2 implies
that, for 0 < a < 1,
oo
{X(t),o < t < 1} = {cy°537irrI/ai(t/i < *), o < t < 1};
for a = 1,
r o °° 2 2 i
{X(t),0 < t < 1} ii-YijiT-hiUi < t)-ptb\l))+pt- In-, 0 < t < 1};
t-7T L—' TV IT J
t=l
and for 1 < a < 2,
oo
{x(t),o< t<\}± [c^Yjai^m < t)-ptt]), o < t < i},
i=l
where {71,72,...} is a sequence of i.i.d. random variables satisfying
p(7i = i) = i-p(7i = -i) = !±£,
{Ti, T2,...} is a sequence of arrival times of a Poisson process with unit arrival
rate and {U\, Ui,...} is a sequence of i.i.d. random variables uniformly distributed
on [0,1]. These three sequences are independent. Finally, the constants b\ are
given by (3.9.1).
This means that we can regard a-stable Levy motion with 0 < a < 1 and
SaS Levy motion with any 0 < a < 2 as pure jump processes. The instants
UiS of the jumps are distributed uniformly over [0,1], the direction of a jump (■ji)
is upward with a probability ^£ and downward with a probability ^f-, and the
height of the jumps viewed in decreasing order is distributed as the —1/a power
of arrival times of Poisson process with unit arrival rate. In particular, T, is
the height of the highest jump, TJ is the height of the second highest jump
and so on. The terms l(Ui < i), i = 1,2,..., indicate which jump sizes have
occurred by time t.
Non-symmetric a-stable L6vy motions with 1 < a < 2 can be regarded as a
pure jump process with a linear deterministic trend.

152 STABLE PROCESSES AND INTEGRALS 3.11
In the symmetric case, we have
Corollary 3.10.4 IfM, in Theorem 3.10.1, is SaS, 0 < a < 2, then for any
feF,
oo
/(/) = (CW£))1/Q X>r-1/'7(V'i), (3.io.2)
where {tui > 1} is an i.i.d. sequence satisfying P(ei = 1) = Pfe = —1) = 1/2,
independent of{Ti,i > 1} and {Vi, i > 1}.
3.11 Condition S
An important modification to the series representation for SaS processes given in
Corollary 3.10.4 is obtained by replacing the Rademacher sequence {e\, €2,...}
by the sequence {d~l G\, d~lG2,...} where G\, G2,... are i.i.d. standard normal
random variables and E\d~ lG\\a = 1. This is, of course, permissible by Theorem
1.4.2. Now, d% = 2Q/27r-'/2r(s±i) by (3.7.2). If Ca is as in (1.2.9) and
C'a =d-"Ca ==2-°/V/2(r(^))"'cQ, (3.11.1)
then
00
Hf) = (CXfiJJ'/^Gir-'^/fK). (3.11.2)
Consider, now, a SaS stochastic process represented as
{X(t) = / ft(x)M(dx), t e T}.
Je
Arguing as in Corollary 3.10.2, we obtain
00
{X(t), t eT} £ {(C'am(E)y/aY/Gir-l/aft(Vi), t er). (3.11.3)
<=i
Now view the series on the right-hand side of (3.11.3) as defined on the
product of two probability spaces: (Q^^Pi) on which the Gaussian sequence
{G\, G2,...} is defined, and (Q2, ^2, Pi) on which the sequences {T\, 1^,...}
and {Vi, V2, ■ • •, } are defined. Therefore, if we "fix" the values of Tj, j —
1,2,..., and Vj, j = 1,2,..., the series on the right-hand side of (3.11.3)
becomes a series of the form {53fei Ai(t)Gi, t € T}, which represents a zero mean
Gaussian process on the probability space (Qj, T\, P\). We obtain, therefore, the
following:

3.11
CONDITION S
153
Proposition 3.11.1 Let M be a SaS random measure on {E,£) with a finite
control measure m. Let ft G La(m), t € T. Then the SaS stochastic process
X(t) = JE ft(x)M(dx), t e T is conditionally centered Gaussian, i.e., it is a
probability mixture of zero mean Gaussian processes.
This statement allows us to relate SaS processes to Gaussian processes whose
structure and properties have been extensively studied.
In Section 3.7, we introduced the sub-Gaussian SaS processes. These are
processes of the form {X(t) — A1/2G(t), t € T} where G(t) is a centered
Gaussian process. It is, therefore, obvious that sub-Gaussian SaS processes are
conditionally centered Gaussian. Proposition 3.11.1 implies that not only sub-
Gaussian SaS processes but also many other SaS processes are conditionally
centered Gaussian; in fact, all SaS processes that can be represented in the form
{X{t), t G T} = | f ft(x)M{dx), teT\, m finite, (3.11.4)
where M is a SaS random measure on (E, £) with a finite control measure m.
But how general is the class of SaS processes {X(t), t G T} that admit the
representation (3.11.4)? This class turns out to be very broad. We will spend the
rest of this section discussing SaS processes that admit representation (3.11.4).
Definition 3.11.2 Let {X(t), t G T} be a stochastic process. We say that it
satisfies Conditions if there is a countable subset To C T such that for every* G T,
X(t) is a limit in probability as n —» oo of sums of the type $Z™=1 ajtnX(tj<n),
for some a,jtn G R, i,-,„ G To, j = 1,..., n, and some n = 1,2, —
Remarks
1. The letter "S" stands for separability.
2. Condition S is equivalent to separability in probability, i.e., to the following:
"There is a countable subset T0 C T such that for every t € T, X(t) =
plimfc_00X(tfc), where tk G T0) k = 1,2,.... " (see Exercise 3.21.)
3. We will see in Chapter 13 that all SaS processes satisfying Condition S do
admit the representation (3.11.4).
We are now ready to present an undoubtedly partial list of SaS processes
that admit the representation (3.11.4). For convenience, we present this list in the
following:

154
STABLE PROCESSES AND INTEGRALS
3.11
Proposition 3.11.3 A SaS process {X(£)-, t G T} admits the integral
representation (3.11.4) if it satisfies at least one of the following conditions:
(i) (T, d) is a separable metric space, and there is a countable subset T\ CT
such that {X(t), t G T\T\} is continuous in probability.
(ii) {X(t), t€T} = {fE gt(x)M'(dx), t € T), where M' is a SaS random
measure on (E' ,£') with a a-finite control measure.
PROOF: We first show that if {X{t), t G T} satisfies Condition (i), then it
satisfies Condition S. Since (T, d) is separable, we can choose a countable set
T2 C T such that for any t G T, there is a sequence {ife}^ in T2 such that
d(tk,t) -> 0 as k —► oo. Since {X(t), t G T\Ti} is continuous in probability,
for every t € T - T,, X(t) is a limit in probability of {X(sk)}, sk G T2, k =
1,2,— Hence, for every t € T, X(t) is a limit in probability of {X(sfc)}
where sk, k = 1,2,..., belongs to the countable set T0 = T\ U T2. The process
{X(t), t € T}, therefore, satisfies Conditions. As noted in the preceding remark,
this implies that the process admits the representation (3.11.4).
Suppose, now, that {X(t), t € X1} satisfies (ii). Let m! be the control measure
of M' and let A\, A2,... be a partition of E' into ^'-measurable sets such that
0 < m'(Ai) < oo, i — 1,2,... . We now apply Proposition 3.5.5 to conclude
that
{X(t), teT}i{J ^ gt(x)r(x)M(dx), i € t},
where, r(x) = (2%m'{Ai))xla if x € Ai, and M is a SaS random measure on
(E',£') whose control measure m(dx) = 2~l{m'(Ai))~lm'{dx), x € Ai, is a
probability measure. This completes the proof. I
The class of SaS processes described in Proposition 3.11.3 is broad enough
to cover virtually every SaS process that we will encounter in this book.
Counterexample. Let X(t), t € [0,1], be i.i.d. (non-degenerate) SaS
random variables. The process {X(t), t € [0,1]} is well defined since its finite-
dimensional distributions exist and form a consistent family. It clearly does not
satisfy Condition S. One can also check directly that it does not have
representation (3.11.4) with finite (or cr-finite) control measure since, if it did, the supports
of ft, t G [0,1], would all be disjoint and also have positive measure, which
is not possible. Nevertheless, this process {X(t), t G [0,1]} is conditionally
centered Gaussian since it can be represented as X(t) = c^A(t)G(t), where
A(t), t G [0,1], are i.i.d. and G(t), t G [0,1], are i.i.d. standard normal random
variables independent of the A(t)s.

3.12
USING A POISSON REPRESENTATION
155
3.12 A fourth definition of stable stochastic integrals
using a Poisson representation
Let M be an a-stable random measure on (E, £) with control measure m (not
necessarily finite) and skewness intensity (3, and let / be any element of the space
F defined in (3.2.1). Our goal is to represent the a-stable stochastic integral
/(/) = fE f(x)M(dx) as an integral with respect to a Poisson random measure
N on £ x Mo, where M0 = K\{0}.
Note that / € F implies that the control measure m is cr-finite on the support
of / because fB \f(x)\am(dx) < co implies m(Ei) < oo where Ei = {x :
2~% < \f{x)\ < 2-l+1}, i — —oo,.... -i-co. We will therefore assume without
loss of generality, that the control measure m is cr-finite. The <7-finiteness of m is
an important requirement in the following construction of the integral.
The heuristic idea is to let
N(dx, du) = number of randomly placed points in dxdu , x g E, u 6 Ro,
(3.12.1)
have a Poisson distribution with a mean n(dx, du) = EN(dx, du) adequately
chosen, and to express M, in the case a < 1, as M(dx) = L uN(dx, du), i.e.,
as the sum of the heights of all the points located within a strip dx in E. The
situation is more delicate when 1 < a < 2 because N has "too many" small jumps
and the integral L uN(dx, du) diverges. It is then necessary to "compensate"
adequately for the jumps.
We start with a formal definition of a Poisson random measure. Let (S, S, n)
be a measure space and let So = {A e 5 : n(A) < oo}.
Definition 3.12.1 A Poisson random measure N on (S, S, n) is an independently
scattered cr-additive set function JV : <S0 —♦ L°(Q) such that, for each set A in So,
the random variable N(A) has a Poisson distribution with mean n(A), i.e.,
P(N(A) = fc)=e-n<A)^^>fc = 0,ll2,...,
n is called the control (or mean) measure of N.
Note that this definition is analogous to the definition of an a-stable random
measure given in Definition 3.3.1. The only difference is that the distribution
is Poisson instead of a-stable. The fact that the Poisson distribution is integer-
valued makes Poisson random measures somewhat more intuitive than their stable
counterparts. Think of a Poisson random measure N as a point process on S: for
each A 6 So, N(A) can be regarded as the (random) number of points belonging
to A (see (3.12.1)), which is why N is also called a counting measure. Then every

156
STABLE PROCESSES AND INTEGRALS
3.12
realization of N is a (non-random) discrete measure on S and the integrals with
respect to N can be defined for each realization. As all the measures involved
are discrete, the integrals, in fact, are sums. Thus, in a sense, the idea of Poisson
representation of stable stochastic integrals has much in common with the series
representation of these integrals developed in Section 3.10.
Set S = E x Rn and choose
n(dx, du) = EN(dx, du) = <
(l+^sHir)^ if«>0,
(I - P(x))m(dx)r$+r ifu<0,
(3.12.2)
where m and /? are, respectively, the control measure and skewness intensity of
the a-stable random measure M.
Let Ei,i=l,2,..., be a partition of E into £-measurable sets of finite
measure m, let 6 > 0 and let (—<5,6)c denote the complement of the interval
(-6,6).
Theorem 3.12.2 (Poisson Representation). The a-stable stochastic integral
1(f) = fB f(x)M(dx) has the representation:
(a) 0 < a < 1 :
/(/)= f2a-1(r(l-a))cos^)~' ° / / f(x)uN(dx,du). (3.12.3)
v 2 / JeJrc
(b) 1 < a < 2 :
x lim Y ( [ f f(x)uN(dx, du) - E / / f(x)uN(dx, du)).
(3.12.4)
(c) a = 1 :
1(f) ^-Mm f]( f f f(x)uN(dx,du)
-2 (ln+ i) J J ^ f(x)p(x)m(dx)\ - ^ f(x)p(x)m(dx) (3.12.5)
where
j f 0 if6>\,
ln+ x = {
b ( m| H6<\,

3.12
USING A POISSON REPRESENTATION
157
and
sinu - ul(u < 1)
f
■ = In -k + /
Jo
du.
o V-
The expressions on the right-hand sides of (3.12.3), (3.12.4) and (3.12.5) converge
a.s.
Remarks
1. As mentioned above, the stochastic integrals with respect to a Poisson
random measure N are denned as sums: for a measurable set A C E x H.0
and a measurable function g : A —» R, we have JAg(x,u)N(dx,du) =
Z)Si ^A({Xi,Ui))g(Xi,Ui) whenever the series converges absolutely,
where {(XitUi), i = 1,2,...} is an enumeration of the points of the
random measure N.
2. If m(E) < co, then set Ex = E and E2 = E3 = •'• • = 0.
3. When 1 < a < 2, the integrals on the right-hand side of (3.12.4) can be
written JE, J,_s S)c f(x)uN(dx, du), where N — N - EN = N -n.
4. When a = 1,
2(ln+-N) / f(x)/3{x)m{dx) = f [ f{x)ul(u < 1) n(dx,du).
V O'.JEi JEiJ{-&,6y
To verify this, use (3.12.2), ln+ | = f£° u l(u < \)u~2du and write
2 [ f(x)(3(x)m(dx) = [ [f(x)(l+/3(x)) + (-f(x))(l-(3(x))}m(dx).
JEi JEi
We cannot replace 2(ln+ \) /£ f(x){3(x)m(dx) on the right-hand side of
(3.12.5) by JEi J,s 6)c f{x)u n(dx, du) because
du
oo.
f°° Z-00 du
/ un(dx,du) — (1 + p{x))m(dx) / u-j
5. To understand heuristically why the Poisson integrals in the theorem are
well defined, recall that the Poisson random measure N(dx, du) has control
measure n{dx,du) = EN(dx,du) < 2m(dx)j^+r. Fix a strip dx and
consider the dependence on u. The mean density of points M_0,_l tends
to infinity as \u\ -* 0. For a > 0, the mean number of points in (-a, a)c,
/, [>an(dx,du) < 4m{dx)f^°u-a-ldu is finite. There are therefore
only finitely many points in the strip {dx, \u\ > a) and the integral on JV

158 STABLE PROCESSES AND INTEGRALS 3.12
with respect to |u| > a is, in fact, a finite sum. Now consider the strip
(dx, \u\ < a).
When a < 1, the expected sum of absolute heights of the points in
(—a,a), f"a\u\n(dx,du) < Am{dx) f°u~adu, is finite. This is why
we can use a Poisson integral with respect to N in (—a, a) when a < l.6
When 1 < a < 2, / "a \u\n(dx, du) is infinite, but f ° u2n(dx, du) <
4m(dx) f£ ui~adu is finite. This is why we can use a Poisson integral
with respect to N = N - EN = N - n in (-a, a) when 1 < a < 2. 1
Observe that when 1 < a < 2, the Poisson representation in (3.12.4)
involves N, even on (—a, a)c. This is valid because
ul{|u| > a}dN = ul{|u| > a}dN - ul{|u| > a}dn
and f, ,c \u\n(dx, du) < Am{dx) f£° u~adu is finite when a > 1.
The proof of the theorem uses the following construction of a Poisson random
measure on E x (6, oo), 6 > 0.
Lemma 3.12.3 Let (E,£,fj.) be a finite measure space, 0 < a < 2 and 6 > 0.
Introduce
1. Z\,Zi,..., E-valued independent random variables with common law
p, = p/p{E).
2. Y\, Y2, ■ ■ ■, real-valued independent positive random variables with
distribution
6a\-a ifX > 6,
P{Yi >X}={ (3.12.6)
1 otherwise.
6If N is a Poisson random measure on (5,S,n), then the Poisson integral Jqg(s)N(ds) is
defined for all measurable functions g satisfying J \g(s)\n(ds) < 00. See Exercise 3.22. By the
same exercise, the characteristic function of J g(s)N(ds) is
EexpliB I g{s)N(ds)\ = expj / (ei8^3' - l)n(ds) j
and|eifl9M-l| <\0g(s)\.
7The Poisson integral J g(s)N(ds) is defined for all g satisfying J g2(s)n(ds) < 00 because
the characteristic function of J g(s)N(ds) is
EexpUe / s(s)iv((is)| =exp| / [eiS»W - 1 - ieg(s)]n(ds)\
andle^W - 1 - i0g(s)\ < \82g2(s). See Exercise 3.23.

3.12
USING A POISSON REPRESENTATION
159
3. D, a Poisson random variable with mean ED = a l<5 an(E).
Assume {Zi},{Yi}, D independent. Then the points (Zj,Yf), j = 1,...,£>,
are a realization of a Poisson random measure N on E xMq with control measure
EN(dz,dy) = ^(dz)^l(y>6).
PROOF: Let A\,..., An be disjoint measurable sets in E x E0 and let
D
iV(^) = £u,(Zfc,Yfc)
fc=i
denote the number of points (Zk,Yk), k — 1,... ,D, in the set Aj. Then N
is a counting measure and N(A\),..., N(An) are independent Poisson random
variables because they correspond to a splitting of the Poisson number of arrivals
D. To compute the control measure of N, note that
EN (A) = (ED)P{(ZUY]) e A}
- a~lS-ap,{E) f l(y> 6)a6ay-(-l+a)p,(dz)dy
J A
= / \(y > 8)y-(l+a^(dz)dy. I
J A
The proof of Theorem 3.12.2 proceeds by splitting E into subspaces Ei with
m(Ei) < oo, defining for each i, two independent Poisson random measures, one
on Ei x (5, oo), the other on Ei x (—oo, —6), and showing that the right-hand
sides in Theorem 3.12.2 converge a.s. as 6 —* 0 to a random variable which has
the same distribution as 1(f).
Proof of Theorem 3.12.2:
(a) 0<q < 1:
Set J(f) ^ JE J9cf(x)uN(dx,du). U(XktUk), k =,0,1,..., is an
enumeration of the points of the Poisson random measure N, then
j(f) = J2u^x^-
fc>0
Our first task is to show that
fe>0

160
STABLE PROCESSES AND INTEGRALS
3.12
It is obviously sufficient to consider the case 0(x) — 1. Then n(dx, du) = 0 for
u < 0 by (3.12.2) and hence Uk > 0. We can write
oo
H(f)= E W(**)l+£ E Uk\f(Xk)\, (3.12.7)
k:(Xk,Uk)eA, m= 1 k:(Xk,Uk)€Bm
where AA = {(x,u) € E x R+ : u|/(z)| > A}, A > 0, and Bj =
A(j+i)-i\Aj-i, j = 1,2, Observe that
n' " '
{Ax) = LJ2m{dx)^
u~(1+a)du
'. / m(efa;) /
JB ./A
+1
oo
A|/(z)|-'
= 2a-'A"a / \f(x)\Qrn{dx)
= C\~a, (3.12.8)
and n{Bj) = C{(j + \)a -ja), where C = 2a~l JE \f(x)\am(dx) < oo. Since
(3.12.8) implies that for almost any realization of N, only finitely many points are
in Ai, the first term on the right-hand side of (3.12.7) is finite with probability 1.
Moreover, if Nj equals the (random) number of points in Bj, then ENj — n(Bj)
and
OO OO -. OO -
^E E Uk\f(Xk)\<E'£1Nj = j:iENj
j=l k:(Xk,Uk)eBj j=l J j=l
°° (n J, 1\° ,-a °°
<cE ~ <^Eja"2<o°
since 0 < a < 1. The double sum in (3.12.7) is therefore finite with probability
1. This proves H(/) < oo a.s. and, thus, that J(f) is well defined in the sense of
a.s. convergence.
Our next step is to prove (3.12.3). Let
Mf)= E u"f(Xk), *= 1,2,... ,
k:XkeBi
where E\,E2,...i$a partition of E into £-measurable sets of finite measure m.
Clearly, J\ (/), Ji{f),. ■. are independent random variables and
OO
i=l

3.12
USING A POISSON REPRESENTATION
161
since Y^Zi \J*(f)\ ^ H(f) < oo. Now choose 6 > 0 and define for each
* = 1,2,...
f J&if) = Ek-^em^s Ukf{Xk),
I (3.12.10)
Then J^if) and J^t{f) are independent random variables and
J<(/) = Hm(J+(/) + J," (/)) a.s. (3.12.11)
o—>u
Now Jg'i(f) is actually a sum over the points of a Poisson random measure on
E x Mo with finite control measure mp(dx)l(x € £i)u~(1+Q)l(ti > S), where
mp{dx) = (1 4-/3(x))m(da:).
Therefore, by Lemma 3.12.3, we can write
^(/) = EW*). (3-12-12)
where {Yj ,12,...} and {Z\, Z2,...} are two independent sequences of i.i.d.
random variables: Yk has the law (3.12.6), Zk has the law mp(dx)lEi{x)/mp(Ei),
and Nfi~t is a Poisson random variable, independent of the sequences {Yk} and
{Zk}, with mean a~16~amp(Ei). Arguing as in the proof of Proposition 1.2.11,
we obtain
£exp{i0J,+(/)} = exp{a-1S-Qmp{Ei)(EeieY'f<-Zl) - 1)}. (3.12.13)
If 4>f denotes the characteristic function of f(Z\),
/•OO
EexpiiOYJiZi)} = / <f>f(e\)a8a\-(a+l)d\
= r aSaX^a+lUX f cifl«l)A(m/,(Ei))-,m^(di)
• = a6a(mp(Ei))-1 [ mp(dx) f° eief^xX^a+1)d\.
Substituting this relation in (3.12.13), we obtain
EexpiieJ+M)} = exp{ f mp(dx) f*'(eiV(,)A - l)A-(Q+1)<w}.
(3.12.14)

162 STABLE PROCESSES AND INTEGRALS 3.12
In the same way,
JSexp{i0J£(/)} = exp{y m.fi(dx) /""'(e-«/(*)* - l)\-^d\],
(3.12.15)
where m-p(dx) = (l-/3(x))m(dx). By the independence of J^(/) and ^"(J),
(3.12.11) and (3.12.9), we obtain
£exp{i0J(/)}
= exp{ I mp{dx) r°(eie^x)x - l)A-(,+0,)dA
+ J m-p(dx) j (e-'^W* - l)A-(,+a>dA}. (3.12.16)
The integrals over A on the right-hand side of (3.12.16) are well known:
/■°°(e±tf/(x)A _ 1)A-(l+a)dA
JO
= -o-'r(l - a) (cos ™)\6f(x)\a (l T t sign (*/(*)) tan ™)
(3.12.17)
(see Feller (1966), p.542, and also Exercise 3.24). Substituting (3.12.17) in
(3.12.16) and using m±p(dx) = (1 ±0(x))m(dx), we conclude Eexp{iOJ(f)}
equals
txp{-2a-lr{l-a)(cos ~) \6\a J \f{x)\a (l-»sign (*/(*)) tan ™)m{dx)}.
Hence, J(/) ~ Sa(<Tf,Pf,0) with
cr? = 2a-'r(l - a)(cos ™) | |/(z)rm(dx),
and Pj defined as in Property 3.2.2. Therefore, by Property 3.2.2,
J(/)i(2a-1r(l-a)cos^)",/(/).
(b) 1 < a < 2 :
Let JfiU) and Jiiiif) be defined as in (3.12.10). We must prove

3.12
USING A POISSON REPRESENTATION
163
where
oo
J(f) = Jim £(J&(/) - EJ&V) + J£(/) - EJfti{fj). (3.12.19)
~~* i=l
Our first task is to show that the right-hand side of (3.12.19) converges a.s.
Observe that the random variables J/j(/), i = 1,2,..., J^i(f), i = 1,2,...,
are independent. Relation (3.12.12) expresses J^(/) in terms of {Yk),{Zk} and
N+{. Since a > 1, EYX = /~ Aa5QA-Q-'dA = a6/(a - 1) < co and
£J+(/) = EN^EYxEf{Zx)
= (a-15-Qm/3(Ei))(Q(Q-l)-'(5)m/3(JBi)-1 / f{x)mp(dx)
JE{
= (a-l)-1<5'-Q / f(x)mf3(dx).
A similar expression holds for EJ^f) with <51-Q replaced by -<51_Q and m/?
by m_/3. Define
and
t=i
We claim that this series converges a.s. Since it is a sum of independent random
variables, we need only to prove convergence in distribution. By (3.12.14) and
(3.12.15),
£exp{i0J^(/)} = exp{ / m±p{dx) f (e±i6i{x)x - \)X^x+a)dX
TiO 7 / f(x)m±0(dx)\
= exp{ J m±0{dx) r°[e±ieKx» - 1 TiOf(x)X}\-^+a)d\y
The exponent in Eexp{i6(JS}i(f) + J^iif))} equals
Ai = [ m0{dx) r[ei9f^)x-\-i6f(x)\}X-^+aU\
+ f m_/5(di) /°°[e-£8/(l)A-H-^/(x)A]A-(1+Q^A,
JEi J6
since the J^(f) are independent random variables.

164
STABLE PROCESSES AND INTEGRALS
3.12
With the changes of variables in the inner integrals, u = A|/(x)| and u =
-A|/(x)|, we obtain
\Ai\ < 2 [ \f(x)\am(dx) f°° \ei0u - 1 - iOu\\u.\-tl+a>du
< C f \f(x)\<*m(dx),
JEi
where C is a finite constant. Since E = jj^i Eu this implies J2il\ l-^l < °°
and, therefore, Js{f) is well defined in the sense of a.s. convergence,
J(f) = \imJs(f)
o—'0
exists in distribution, and
J5exp{i0J(/)} = exp{ f mp(dx) j [eie^x - 1 - ^/(x)A]A"(1+Q)dA
+ f m-p(dx) f [e~ief^x - 1 + i0/(x)A]A-('+a><iA}. (3.12.20)
To see that J(f) is defined a.s., note that the summands in
oo
?(f) = Z)(^-i(/) - £-*+'(/)) + Ml) (3-12.21)
are independent random variables and hence the convergence in distribution of
the sum implies a.s. convergence.
It remains to prove (3.12.18). To do this, we need to compute the integrals in
(3.12.20). An integration by parts of (3.12.17) (see Exercise 3.24) gives
/•OO
/ {e±iB^x)x - 1 T iOf{x)\)\-(l+a)d\
Jo
= -W^^T) (-cos t) 0 T •sign mx)) tan T )•
Substituting this relation in (3.12.20) and using m±p(dx) = (1 ± f3(x))m(dx)
yields
£exp{i0J(/)} =
exp{2^^(cos^)|0r^|/(x)r(l-isign(0/(x))tan^)m(dx)}.
Hence, J(f) ~ Sa(af,0f,O) with af = 2g|=2i(-cos *?) fB \f(x)\am(dx)
and Pf is defined as in Property 3.2.2. Therefore Property 3.2.2 implies Relation
(3.12.18).

3.12 USING A POISSON REPRESENTATION 165
(c) a = 1 :
Because the main steps are the same as in the case 1 < a < 2, we consider
only differences in computation. Once more, let J/^/) and J^if) be defined as
in (3.12.10). We must prove
/(/) = ~J(f) ~ ~ I f(x)0(x)m{dx), (3.12.22)
where this time
J(/) = lim V(j+(/) + Jfti(f) - 2(ln+ -A / f(x)P(x)m(dx)).
(3.12.23)
Define
JtiU) = ^(/) T (ln+ r1) / /(i)(l ± p(x))m(dx)
JEi
and J^(/) = X!)i=i(J«i(/) + J(i~i(f))- We firstly prove that this sum converges
in distribution and, hence a.s. We have
EexpiiOJ^if)} = exp{ f (1 ± /3(x))m(dz) /" (e±i9/(x^ - l)A~2dA
Ti^ln+r1) /" /(x)(l±/?(x))m(da:)}
= exp| /" (1 ± p{x))m{dx) f [e±i9/^ - 1 =F i0/(z)Al(A < l)]A~2dA}
since ln+(5_1 = /~A_11(A < l)dA. The exponent in £exp{i0(J^(/) +
/ m(dx) fV9^> + e-i9'W - 2)A-2dA
JBi JS
+ f P{x)m{dx) r[eie^x)X-e-i9f^)x-2i9f(x)Xl(X<l)}\-2d\.
JEi J 6
This can be expressed as M + Bi, where
At = 2 f m{dx) f [cos(9f(x)X) - l]X~2dX,
JEi JS
Bi = 2i [ (3(x)m(dx) [ [sin(0/(x)A) - 0/(x)Al(A < l)]A~2dA.
JEi JS

166 STABLE PROCESSES AND INTEGRALS 3.12
Now with the change of variables u = \8f(x)\\ in the inner integral,
\Ai\<2\6\ I \f{x)\m{dx) I \\ - cosu\u~2du,
JEi JO
\Bi\<2\6\ [ \f(x)/3(x)\m(dx)L(f(x)),
JEi
where, for 6 < 1,
L{f(x)) = / | sinu-ul(u < \)\u~2du
Jo
/•OO
+ / u|l(u< l)-l(u< \9f(x)\)\u-2du
Jo
[ u~ldu <C+|ln|/(s)||,
J\ef(x)i
= C+' '
' /|«/(x)|
for some C < oo, so that
|£i|<2C|0| / \f(x)\m(dx) + 2\6\ [ \f(x)0(x)\n\f(x)\\m(dx).
JEi JEi
E = USi E and (3-2-1) impty E,~i(l^i| + l-B.I) < oo. Therefore J4(/) is
well defined in the sense of a.s. convergence, J(f) — \ims-,o Js(f) exists in
distribution and
£exp{i0J(/)}
= expj-2 J m(dx) f [1 - cos(0/(x)A)]A_2dA
+ 2i f 0(x)m(dx) f [sin(0/(z)A) - 0/(x)Al(A < l)]A-2cL\}.
(3.12.24)
Since (3.12.21) holds also when a = 1, we conclude that J(f) is defined a.s.. It
remains to evaluate the integrals in (3.12.24). We have
/•OO /»00
/ [1 - cos(0/(z)A)]A-2dA = |0/(x)| / (1 - cosu)u-2du = -?\0f(x)\
Jo Jo 2
and
/•OO
/ [sin(0/(x)A) - 0/(x)Al(A < \)}\-2d\
Jo
/•OO
= 9f{x) / [sinu - ul(u < \9f{x)\)]u-2du
Jo

3.13 EXERCISES 167
' [sinu - ul(u < \)\u~2du
o
/•OO
+ / vrx[\{u < 1) - \{u < \6f(x)\)]du
Jo
= 9f{x) (a+ f u^du) = 0/(x)(a - In |0/(x)|),
V J\9f(x)\ '
where
/■CO
a—\ [sinu — ul(u < \)]u~2du.
Jo
Jo
Substituting the preceding relations in (3.12.24) yields
£exp{i0-J(/)} = £exp{i0j(^-) J
= exp|-|6»| J |/(x)|m(dx) + -tela j f(x)f3{x)m(dx)
-\n\9\ [ f{x)P{x)m(dx)- f /(x)/3(x)ln|^/(x)|m(dx)]}
= exp{-a/|0|(l + i{3f- In |0|) + i/i/fl}
where 07 = JE \f(x)\m(dx). 0j is defined as in Property 3.2.2 and
Hf = -{a + lnrc) / f(x)0(x)m(dx) / /(x)/3(x)ln|/(x)|m(dx).
it Je * Je
Hence, £./(/) ~ Si(aflf3f,nf). Property 3.2.2 implies Relation (3.12.22). I
3.13 Exercises
Exercise 3.1 Let F be defined as in (3.2.1). Prove that / € F implies that the
support of / is contained in a region of E where m is cr-finite.
Hint: Integrate |/| over2-n < f < 2-<-n+l\
Exercise 3.2 Show that the stochastic process {M(A),A € So} with finite-
dimensional distributions given by (3.3.1) satisfies
d d
for any disjoint A\,..., Ad in So. (This is finite additivity for a random measure.)
Exercise 3.3 Show that the set of function F defined in (3.2.1) is a linear space.

168 STABLE PROCESSES AND INTEGRALS 3.13
Exercise 3.4 Prove Proposition 3.5.2 using Definition 2.7.3.
Exercise 3.5 Find square integrable functions f\ and fa such that both (3.5.5)
and (3.5.6) hold. (This illustrates that Theorem 3.5.3 is false in the Gaussian case
a = 2.)
Exercise 3.6 Let M be a SaS random measure on (E, S) with control measure
m. Show that there is an isometry between the space of all SaS random variables
of the form
X = / f(x)M(dx)
Je
equipped with the covariation norm and La(E, £, m).
Exercise 3.7 Let Xi, X2 be jointly a-stable random variables in R2 with a^l
and let f\ and /2 be the functions that appear in their integral representation. What
conditions must f\ and fo satisfy if X\ is SaS, X2 is SaS but (Xi, X2) is not
SaS?
Exercise 3.8 Let {X2(t), — 00 < t < 00} be the reverse Ornstein-Uhlenbeck
process. Verify that the spectral measure T2 of the vector (X2(s), .X^i)), s < t,
has the form given in Example 3.6.4.
Exercise 3.9 Prove that the linear fractional stable motion in Example 3.6.5 is
self-similar and has stationary increments. To prove self-similarity, use the change
of variables result given in Proposition 3.5.5.
Exercise 3.10 Show that the log-fractional stable motion given in Example 3.6.6
is well defined, is self-similar with H = 1/a and has stationary increments.
Exercise 3.11 Prove that when a = 2, the log-fractional stable motion (3.6.5) has
the same autocovariance function as the well-balanced fractional stable motion
(3.6.4). The process with a = 2 is called fractional Brownian motion.
Exercise 3.12 Show that (da)a = E\Z\a = 2Q/27r-'/2r(^±i) where Z ~
N(0,1).
Exercise 3.13 Say that a SaS random vector X is sub-sub-stable if, for some
0 < a < a' < a" < 2, it is sub-stable, i.e., X = A1/"' Y, and if Y is also
sub-stable, i.e., Y = Al'a Z. Here A ~ Sa>/a(aA, 1,0) is independent of
Y ~ Sa'S and B ~ Sa"/a,(aB, 1,0) is independent of Z ~ Sa"S.
(1) Show that a sub-sub-stable random vector is sub-stable.
(2) Express oq as a function of a a and ob-
Hint for (I): Show that there is a random variable C ~ SQ» ,a (ac > 110)
independent of Z such that X = C"/Q"z. Take Cl'a" = Ax/a' Bx'a" and verify that C
has the required distribution.

3.13
EXERCISES
169
Exercise 3.14 Prove Theorem 3.10.1.
Exercise 3.15 Show that a SaS random vector X in Rd, 0 < a < 2, can be
represented as
oo
* = 2>rrI/aw„
i=\
where {ej}, {r*}, {Wj} are independent sequences, the e^s and TjS are defined
as in Theorem 3.10.1 and the Wj = (W^\..., W^d))s are i.i.d. vectors in Rd
satisfying B\\Wi\\a < oo or, equivalently, E\W^\a < oo, j = 1,...,d.
Exercise 3.16 Let{ri,r2,...} be a sequence of arrival times of a Poisson process
with unit rate and let {Gi(t),t € T}, i = 1,2,..., be i.i.d. Gaussian processes
with mean zero and independent of {Ti, T2,...}. Prove that the series
oo
x(t) = 2-°i2cxJad-a J2 Gi(t)r;l/a
converges a.s. for each t € T and {X(t), t 6 T} is a sub-Gaussian process. (The
constant Ca is defined in (1.2.9) and da = (E\N(0, l)|a)1/Q.)
Exercise 3.17 A subset C of a metric space (T, d) is said to be separable with
respect to d-convergence if there is a countable subset Co C C such that for every
u £ C, there is a sequence {ufc}£L, in Co such that limfc-voo d(u, Uk) = 0.
Prove that a subset of a separable set is again separable.
Exercise 3.18 Show that every non-random function {^(t), t € T}, regarded as
a (degenerate) stochastic process, satisfies Condition S.
Hint: The subset {y = <f>(t), t £ T} of 1R is separable.
Exercise 3.19 Show that a stochastic process {X(t), t € T} satisfies Condition
S if and only if the linear span Lx of the process is separable with respect to
convergence in probability. The linear span Lx is the linear space of the type
E"=i ajX(tj), a, € M, tj e T, j = 1,2,..., n, n = 1,2,....
Exercise 3.20 Let {X(t), t e T} and {Y(t), t € T} be two stochastic processes
defined on the same probability space and satisfying Condition S. Then {X(t) +
Y(t), t£T} also satisfies Condition S.
Exercise 3.21 Use Exercises 3.19 and 3.17 to show that Condition S is equivalent
to the following, apparently stronger condition, that {X(t), t € T} is separable
in probability, i.e., that there is a countable set T0 C T such that for every t € T,
there is a sequence {ik}^ in T0 such that X(t) = plimk_00X(tk).

170
STABLE PROCESSES AND INTEGRALS
3.13
Exercise 3.22 Let AT be a Poisson random measure on (S, S, n), and let g be a
measurable function: S —> R satisfying fs \g(s)\n(ds) < oo. Then the Poisson
stochastic integral Js g(s)N(ds) converges a.s., and its characteristic function is
given by
Eexp{iB J g{s)N{ds)} = exp{ f (eifl»W - l)n(ds)}.
Hint: The control measure n must be a-finite on the support of g.
Partition this support into sets At satisfying n(Ai) < oo, so that J2k>o Ip(^0I =
Z)Si Z)fe>o 1^i(-^fc)|p(-X'fc)|, with {Xfe} being a realization of N. For each
i = 1,2,..., devise a construction of N restricted to Ai in the spirit of Lemma
3.12.3, and use an argument similar to the one used to prove that H(f) < oo a.s.
in (3.12.7).
Exercise 3.23 Let N be as in Exercise 3.22, and let g this time satisfy
Is \9(s)\2n(ds) < oo. Then
J g(s)N(ds) -f2(JE 9(s)N(ds) ~eJ 9{s)N{dsj)
is well defined in the sense of a.s. convergence. Here, {Ei} is an arbitrary partition
of S into sets of finite measure n. Show that fs g(s)N(ds) does not depend on
the chosen partition.
Hint: Argue as in Exercise 3.22 and use the fact that a series of independent
random variables converges a.s. if and only if it converges in distribution.
Exercise 3.24 (1) Prove that for 0 < a < 1,
J TOO
' (eiaA - l)\~V+a)d\
o
= -|a|Qa-'r(l-a)cos^(l-z(signa)tan^). (3.13.1)
Hint: Suppose first a > 0 and view the integral as the limit as v tends
to 0 of j0oo(eiaA-"A - l)A_(1+Q)dA. After integration by parts, obtain
(ia — u)a~l J?° eiaXe~"x\~ad\, an integral proportional to the characteristic
function of a gamma distribution. Since
Jo
oo ,
T(t) \ 1 — iau
the right-hand side of (3.13.1) equals -(-ia)aa-1r(l - a). If a < 0, take the
complex conjugate. Question: Why introduce vl

3.13
EXERCISES
171
(2) Prove that for 1 < a < 2,
r(e^-l-ioA)A-('+a)dA = |a|-^l2|co«^(l-i(«gnfl)tan^).
Jo oc{a - 1) 2 V 2 /
(3.13.2)
Hint: Suppose first a > 0, integrate by parts to obtain aaia~l f0°°(eiu — \)u~adu
and use (3.13.1) to obtain
)]
—aia
(a-l)-lr(l-(a-l))cos
a T(2 — a) ira
7r(a — 1)
. ,, cos —- (1 — i tan ■
a(a- 1) 2
(
(l-.taiT).
1 — i tan
7r(a - 1)
If a < 0, take the complex conjugate.

Chapter 4
Dependence structures of
multivariate stable
distributions
In this chapter we describe a number of properties of multivariate stable
distributions. We start with conditional expectations. Let (Xi,X2) be symmetric
a-stable with 1 < a < 2. We show in Section 4.1 that, as in the Gaussian case,
E(X2\X\) is linear in X\. In fact, E(X2\Xi) = cX\, and the coefficient c equals
the normalized covariation of X2 on X\. (See Chapter 5 for extensions to a < 1
and to the case where (X\, X2) is skewed a-stable).
The analogy to the Gaussian case breaks down if one conditions on more than
one variable. Suppose that {X\,..., Xn) is symmetric a-stable, 1 < a < 2. Then
E(Xn\X\,..., Xn-i) is usually not linear. It is linear in some special cases, for
example when X\,..., Xn_ \ are independent. If (X{,..., Xn) is sub-Gaussian,
then, as in the Gaussian case, any random variable in the linear span ofX\,..., Xn
is linear.
If (X\,..., Xn) is Gaussian, then the conditional laws are also Gaussian.
Unfortunately, very little is known about the conditional laws £{Xn\X\,..., X„_i}
when X\,..., Xn are jointly a-stable. The conditional law is, in general, not
a-stable and it is, in general, not even symmetric around the conditional mean.
We provide in Section 4.2 a necessary and sufficient condition for the symmetry of
£{X2\X\} around the conditional mean cX\. In the special case where the
conditional law C(X2\X\) is a-stable, symmetry around the conditional mean cX\
occurs if and only if Xi and X2 are linearly dependent, i.e., if X2 = cX\ + Y,
where Y is SaS and independent of X\ (Section 4.3).
In Section 4.4 we analyze the asymptotic behavior of order statistics. Suppose

174
DEPENDENCE STRUCTURES
4.1
that X\,..., Xn are jointly a-stable, 0 < a < 2. We show that the tail
P{X(fc) > A}, k = 1,..., n, of the distribution of the order statistics X^
is always asymptotically proportional to X~a as A —> oo, i.e., it has the same type
of asymptotic behavior as the tail of any of the Xj s, j — 1,..., n. The coefficients
of proportionality, however, depend on the joint distribution.
In Section 4.5 we investigate the existence of joint moments of a-stable random
variables X\,..., Xn, 0 < a < 2. For what (non-negative) exponents Pi,...,pn
is the joint moment E\X\ |Pl ... |^n|p" finite? If the XjS are independent, then,
necessarily, pj < a for all j. If the XjS are all equal, then pi -I \-pn < a. We
give a necessary and sufficient condition on pi,..., pn for E\Xi |Pl ... \Xn\Pn to
be finite when p\,..., pn are arbitrary non-negative numbers.
Association is a dependence structure that has been widely used in the finite
variance case. Jointly normal random variables, for example, are associated if
and only if their correlations are all non-negative. The corresponding result in the
a-stable case, 0 < a < 2, is given in Section 4.6.
The codifference between two stable random variables was defined in Chapter
2. We listed there some of its basic properties and noted in particular that the
codifference is identical to the covariance when a = 2. In Section 4.7, we
consider a stationary SaS process X = {X(t), t € R} with index 0 < a < 2
and study the codifference between X(0) and X(t). We investigate the asymptotic
behavior of this codifference as the lag t tends to infinity. The codifference always
tends to zero when the process X is a moving average. For example, it tends to
zero exponentially fast if the process is SaS Omstein-UWenbeck.
The mean number of times that a stationary sub-Gaussian process crosses a
given level u is computed in Section 4.8. It decreases like u~a as u —» oo.
4.1 Linear regression
The regression of a random variable X2 on a random variable X\ is linear if there
exists a constant c such that E{X2\X\) = cX\ a.s. The following lemma provides
a criterion for linearity in terms of the joint characteristic function </>(#i,02) =
Ecxp{i(6iXi + 82X2)} of Xi and X2.
Lemma 4.1.1 Suppose that E\Xi\ < 00 and E\X2\ < 00. Then E(X2\Xi) —
cX\ a.s. is equivalent to
d<fi(eu h) = d<j>(0\,62)
d62 d2=0 ° d9x
Proof: Setting 4>(fiue2) = £exp{i(^,X, +^^2)} in (4.1.1) yields
(4.1.1)

4.1
LINEAR REGRESSION
175
EX2eie<x> =EcXieie'x'
for all real 8\, i.e.,
/oo />oo
/ (cxl-x2)eiex<dF(xu<x2) = 0 (4.1.2)
-oo J—oo
for all real 6, where F is the distribution function of the vector (X\, X2). The
left-hand side of (4.1.2), which we denote fi(0), can be regarded as the Fourier
transform
/oo
ei9xn{dx), -oo < 6 < oo,
■oo
of the signed measure ponR1 given by
H{B)= [ [ (ex, - x2)dF(xux2). (4.1.3)
JB J-oo
Relation (4.1.2) implies fi = 0. To verify this, we can appeal to the uniqueness
of characteristic functions. Indeed, let us write /i = p,\ — \x2 where ^i and \x2
are two non-negative finite measures whose Fourier transforms are /2i and fl2,
respectively. Relation (4.1.2) implies /I| — Jl2 — /2 = 0, i.e. fi\ = ji2. Now,
either fi\ = [i2 = 0 or neither pi nor \i2 is a zero measure. In the latter case,
dividing [i\ and \i2 by the constant a*i(R') = Mi(0) = £2(0) = M2(^') converts
them to probability measures and we have [i\ = fi2 by the uniqueness theorem
for characteristic functions. Hence \x = 0.
Since for any A € cr(Xi),
[ E{cXx - X2\Xx)dP = / (cX, - X2)dP,
J A J A
the condition fi = 0 is equivalent to E(cXi — X2|Xi) = 0 a.s., i.e., E(X2\Xi) =
cX\ a.s. I
As is well known, the regression is linear when X\ and X2 are jointly Gaussian
mean zero random variables, and in that case c = EX2X\/EX]. Now suppose
that X\ and X2 are jointly SaS with a > 1. (The assumption a > 1 ensures
that expectations exist.) The following theorem shows that the regression is still
linear.
Theorem 4.1.2 Let X\ and X2 be two jointly SaS random variables, with 1 <
a < 2. Then
E(X2\Xl)=[X^X'}aXl a.s. (4.1.4)

176 DEPENDENCE STRUCTURES 4.1
PROOF: It is sufficient to verify that (4.1.1) holds with c = [X2,X^a/WXi \\a.
The joint characteristic function of (XUX2) is <t>(0u62) = exp{-<rQ(0i,02)},
where
<ra(91,e2) = [ \elSl+e2s2\ar(ds)
and where T is the spectral measure. By (2.7.2),
daa(9u92)
89,
and
daa(6ue2)
d92
= a / Sl(^Sl)<a-'>r(ds)
= a f s2(9,Sl)<a-l>r(ds)
e2=o JSi
- a9
<Q-1>
[X2,X
!••
Setting
we obtain
c~ II*. Us '
daa(9u92)
d92
= c
3<7Q(0,,02)
0j=O
36>,
e2=o
and, hence, (4.1.1). 1
What about multiple regression? Whereas multiple regression is always linear
in the Gaussian case, this is not so in the SaS, a < 2 case. Necessary and
sufficient conditions for linearity are known, but they are complicated. (See
Miller (1978).) The necessary conditions for linearity given in the following
corollary can be used to construct counterexamples.
Corollary 4.1.3 Let X\, X2,..., Xn be jointly SaS with \ < a < 2. If
E[Xn\Xi,---,Xn-i)=alXi+--- + an-lXn-i a.s., (4.1.5)
then the coefficients a\,..., an-\ satisfy the system of linear equations
/ J Qi[Xi, Xj\a — [Xn, Xj\a , j — 1,..., n — 1.
(4.1.6)
i=1
Moreover, if n = 3 and if Xx and X2 are linearly independent, then the system
(4.1.6) uniquely determines the coefficients.

4.1
LINEAR REGRESSION
177
PROOF: Set dj = [Xi,Xj]a and suppose that (4.1.5) holds. Then, for any
j = l,...,n- 1,
^Xj = E(Xn\Xj)
c
n
= E[E{Xn\X{ + ... + Xj + ~- + Xn^)\Xj}
= E{aiXX + ■■■ + GjXj + ■■• + On-iXn-ilXj)
n-1
= ajXi+^2aiE(Xi\Xj)
i=l
= ajXj+Y^a^Xj
1 ^
== y^&iCijsi.j)
establishing (4.1.6).
When n = 3, the system has a unique solution unless cuC22 = C12C21, i.e.,
unless
/ |si|<T(ds) / \32\aT{ds) = / s,s<a-l>r(ds) / s2sf°-l>r{ds).
JSi JS2 J Sz J St
(4.1.7)
Applying the Holder inequality to the right-hand side of (4.1.7) yields the left-hand
side. This means that (4.1.7) is equivalent to s\ = Xs2 T-a.e. for some real A, i.e.,
to X! = XX2 a.s. I
Remark. When n > 3, linear independence of X\,X2,... ,Xn-\ is obviously
a necessary condition for (4.1.6) to have a unique solution. The converse is true
when n = 3 (Corollary 4.1.3), but is not true when n > 3 (Exercise 4.3).
Example 4.1.4 A SaS law with non-linear multiple regression
Let X\, X2, X->, be three i.i.d. SaS random variables with 1 < a < 2. If
y, = x,+x2, y2 = xk + xz, y3 = x,,
then E(Yi\Yi, Y2) is not linear in Y\ and Y2. To show this, we will use Corollary
4.1.3 and the fact that the covariation is not linear in its second argument.
Suppose, to the contrary, that E{Y^\YX, Y2) = a\Y\ + C12Y2 a.s.. Then, for any
0>O,
[Y3,Yi + 0Y2]a , flVx
m+ms(l+ 2)

178 DEPENDENCE STRUCTURES 4.1
= E(Y3\YX+6Y2)
= E[E(Y3\YuY2)\Yi+6Y2}
= aiE(Yi \YX + 6Y2) + a2E(Y2\Yx + 6Y2)
= (ax[Yx,Yx + 6Y2}a + a2[Y2,Yx + 9Y2]a)-. Yl + 6Yl
\Y+6Y2\\^
so that
PS, y + eY2]a = a, [y, y + 0y2]Q + a2[y2, y, + 0y2]Q.
Setting Qj = [Yi, Yj]a and replacing the coefficients ax and a2 with the solution
to the system of equations in Corollary 4.1.3, yields
[y3,y.+0y2]a
= (C31C22 - c2lc32)[y, y + 6Y2)a + (C11C32 - c3ici2)[y2, y + eY2\a
0\\C22 - C21C12
All these covariations can be simplified; for example,
[YuYx+eY2]a
= {xx+x2,xx+x2 + e(xx+Xi)}a
= [Xu (1 + 6)XX + X2 + 0X3]a + [X2, (1 + 6)XX +X2+ 6X3]a
= (l+e)Q~iaa+aa,
where aa = [Xi,Xi]a = ||Xi||£, i = 1,2,3. After simplification, one obtains
(1 + 0)a-' = |(1 + O*-1 + 2(1 + 0)Q-')
or (1 + 6)a-' = 1 + 6a-', which holds for any 0 > 0 if and only if a = 2. Hence
£,(y3|y, y2) is not linear in y and y2 when a < 2.
The following corollary shows that E(Xn\Xx,... ,Xn^x) is linear if the
random variables Xx,..., Xn-X are independent.
Corollary 4.1.5 Let Xx,..., Xn- x, Xn be jointly SaS with n > 2 and 1 < a <
2, and suppose that Xx,..., Xn-X are independent. Then
n-l
E(Xn\Xx,..., A"n_i) = yckXk
[Xn,Xk]t.
Jt=i
where
n«

4.1
LINEAR REGRESSION
179
PROOF: We can assume by the representation theorem 3.5.6 that
Xk = fk(x)M(dx), k=l,...,n,
JE
where M is a SaS random measure with control measure m and fk S La{E, m).
Moreover, since X\,..., Xn- \ are independent, we can assume that the functions
/i,..., /„_i have disjoint supports (see Theorem 3.5.3). Therefore there are
disjoint sets A\,... ,^4n-i such that fk (x) = 0 for any a; £ Ak, k = l,...,n-l.
On one hand, for any k = 1,..., n — 1,
E( / fn(x)M(dx)
lAk
X\,. ..,Xn-\
(x)M(dx)
j = l,...,n- 1
= E[ f fn(x)M(dx) f fj(x)M{dx),
\JAk JAj
= EI fn(x)M(dx) / fk(x)M(dx) ) (independence)
= ck fk{x)M(dx) (Theorem 4.1.2)
JAk
'A
= ckXk,
where, by Proposition 3.5.2,
Cfc
1
11-^*1
1
[Xn, Xk\g
\\xk\\«
- f fn(x)fk(x)<a-l>m(dx)
a JAk
- [ fn(x)fk(x)<a-i>m(dx)
a JE
On the other hand, /(un-i Ak)c fn(x)M(dx) is independent of X\,..., Xn-\ and
therefore
E
Uu;:;
Aky
fn(x)M(dx)
X\,... ,Xn~\
E f fn(x)M(dx)
J(\ \n-'Ak)°
'tu::>)
0.
Hence
E(Xn\X\,... ,Xn-\)

180
DEPENDENCE STRUCTURES
4.1
= E (£ / Mx)M(dx) + f fn(x)M(dx)
\ fr^ J a* J(\ r-' A.v
<u;:>>«
X\,..., Xn.
n-\
= Y,CkXk
*:=i
We now want to characterize those SaS vectors having linear spans where
all multiple regressions are linear. Let X = (X\ ,X2,..., Xn) be a SaS random
vector with 1 < a < 2 and let sp(X) denote the linear span of X, i.e., the vector
space of all finite linear combinations ££_, A^Xfc, —oo < Aj,..., Xn < oo.
Definition 4.1.6 The vector X has the multiple regression property if for all
random variables Yi, Yi,..., Yn in sp(X),
E(Yn\Yu...,y„_i) € sp{(Y,,..., Yn_,)}.
It is well known that every Gaussian vector has the multiple regression
property. What about SaS random vectors with 1 < a < 2? Theorem 4.1.2 shows that
the property holds when dim(sp(X)) = 2. But, as we shall see, only sub-Gaussian
vectors have the property when dim(sp(X)) > 3.
Proposition 4.1.7 Let X be a SaS random vector with 1 < a < 2 and suppose
that dim(sp(X)) > 3. Then the following three statements are equivalent:
(i) X has the multiple regression property,
(ii) X is sub-Gaussian,
(Hi) James orthogonality in sp(X) is additive on the left, i.e.,
Y,, Y2, Y3 € sp(X), [Y3, Vila = 0, [y3, Y2)a =0 =» [Y3,Yi + Y2)a - 0.
PROOF: The equivalence (ii)-^(iii) follows from Proposition 2.9.3. To prove (i)
=*> (iii), suppose (i) and let Yj, Y2,Y3 be elements of sp(X), satisfying
[Y3,Y1]Q = [y3,Y2]a = 0.
Obviously, [Y3, Yj + Y2]a — 0 if Y\ and Y2 are a multiple of each other. If Y\ and
Y2 are linearly independent,
= £(E(Y3|Y"1,Y2)|Y1+Y2)
= £(a,yl + a2Y2|yl + y2)
= o,

4.2
CONDITIONAL LAWS
181
since, by Corollary 4.1.3, ax = a2 = 0. Hence we obtain again [Y3, Yx +Y2]a = 0.
We now prove that (ii)=>(i). Let Yx,... ,Yn be in sp(X). Since X is sub-
Gaussian, so is (Yx,...,Yn), i.e., Yj — Ai/2Gj, j = l,...,n, where A ~
Sa/2(a, 1,0) and (Gx,..., Gn) is a mean zero Gaussian vector. Then
E(Yn\Yx,...,Yn-X)
= E(Ai'2Gn\Al/2Gu...,Ai'2Gn-l)
= E[[E(Al/2Gn\A,Gx,...,Gn-X)\Al/2Gx,...,Al'2Gn-X)
= E{A"2E{Gn\Gx,...,Gn-X)\Axl2Gu...,Ax>2Gn_x)
n-l
= £(E ^V^ Al/2G< | AW2G» ■ ■ ■^1/2Gn-l)
i=l *
_^Cov(Gi,C7n) Ail2ri
~ ^ VzxGi '
t=i
_^Cov(G,,Gn)
~ Z^ var d u
i=i
establishing (i). I
4.2 Conditional laws that are symmetric around the
conditional mean
We saw in Theorem 4.1.2 that for pairs of jointly SaS random variables, a > 1,
the conditional expectation is linear and behaves as in the Gaussian case. However,
many properties of conditional distributions for pairs of Gaussian random variables
do not extend to the stable case, e.g., symmetry. When (Xi,Xi) are jointly
Gaussian with zero mean, the conditional law X2\XX is symmetric around the
conditional mean because E(X2\XX) = cXx and X2\XX ~ N(cXx, ■). We will
see that when {Xx, X2) are jointly SaS, 1 < a < 2, the conditional law X2\XX
is not always symmetric around the conditional mean.
Suppose (XX,X2) jointly SaS, 1 < a < 2. We know from Theorem 4.1.2
that there is a constant c such that E(X2 \ X\) = cXx. If the conditional distribution
of X2I-X1 is symmetric, it must necessarily be symmetric around its conditional
mean cX\, i.e., for every Borel set B2,
P{X2 - cXx € B2\XX) = P{cXx -X2e B2\XX) a.s. (4.2.1)
Integrating this relation over the event {Xx € Bx} yields
P(X, 6 Bx, X2 - cXx e Bi) = P{XX e Bx, cXx - X2 € B2) (4.2.2)

182 DEPENDENCE STRUCTURES 4.2
for all Borel sets B\ and B2. Of course, (4.2.2) also implies (4.2.1). Therefore
X2\X\ is conditionally symmetric (around its conditional mean cX\) if and only
if
(XuX2-cXl) = (XucXl-X2)
and hence (Exercise 4.8) if and only if
(Xi,X2)±(Xi,2cX1-X2).
(4.2.3)
This last expression states that the distribution of the (column) vector (X\, X%) is
invariant under the transformation
1 0
2c -1
It also shows that (4.2.1) holds if and only if the spectral measures of the SaS
random vectors (X\, X2) and (X\, 2cX\ —X2) coincide. This last criterion would
enable us, for example, to verify that a SaS random vector (X\, X2) with spectral
measure
T-i
'y/2 v/2>
V2 V2>
1
+2
*((t-t))+«((-t--t))
(4.2.4)
satisfies (4.2.3), whereas a SaS random vector (Xi, X2) with the spectral measure
'\/2 y/2\\ , c(( V2 >/2\\~
2 ))
.((£.£))+.((-:
(4.2.5)
does not (see Exercise 4.9). The criterion, unfortunately, is not particularly easy
to visualize because it involves a linear transformation of T on the circle 52-
Theorem 4.2.2 below gives a more natural criterion, involving the translation of a
measure on the real line.
We start with an alternative representation for the joint characteristic function
of(XuX2).
Lemma 4.2.1 LetX\ andX2 be jointly SaS random variables, 0 < a < 2. Then
there is a unique constant b and measure a on K1, satisfying J_oo \t\aa(dt) < 00,
such that
Eexpfi^X, + 62X2)} = exp{- f |0, + 92t\aa{dt) - b\92\a\. (4.2.6)

4.2
CONDITIONAL LAWS
183
Proof: By Theorem 2.4.3,
Eexp{i(9iXi+02X2)}
= exP{- J \elSl + e2s2\ar{ds)}
= exP{- J t \e]Sl + 92s2\ar(ds) - |02nr((o, -l)) + r((o, 1))}}
){- / 6l+e2^aml{ds)-b\92\a\,
L Js' si '
= exp<
where S'2 = 52\{(0, -1), (0,1)}, mx{ds) = |si|Qr(cfe) and b = r((0,~l)) +
r((0,1)) = 2r((0,1)). The transformation h(si,s2) = s2/si from S{ to E1
induces the measure a = ft(mi) = m\ o h~x on R1. Setting t = s2/si yields
(4.2.6). Note that
/ |t|a<r(dt) < f \s2\aT(ds) < f T(ds)
< 00.
The representation (4.2.6) is uniquely characterized by the pair (a, b) because the
spectral measure T is unique and can be recovered from (a, b). I
Our criterion will now be stated in terms of the representation (4.2.6), i.e. in
terms of a (and b).
Theorem 4.2.2 A jointly SaS pair of random variables X\, X2,1 < a < 2, has
a symmetric conditional law X2\X\, i.e., it satisfies (4.2.3) for some c, if and only
if its representation (4.2.6) involves a measure a which is symmetric (around c).
PROOF: Using the representation (4.2.6) we obtain
Eex.p{H,6iX1+02X2)}
= Eexpi{9iX{ + 92(2cXi - X2)}
= £exp{i(0, +2c92)Xi - 92X2}
= exp{- f |0! + 2c<92 - 92t\aa{dt) - b\92\a\
^ J —CO
= exp{- r \9i+62{2c-t)\aa{dt)-b\92\a\
^ J—oo
= exp{- I" |0, + 92s\aal(ds) - b\92\a\
^ J—oo
by settings = h(t) = 2c-i and letting o x be the induced measure h(a) = ooh~x.
Hence {X\, X2) has also the representation (4.2.6) but with measure v\. Since the

184 DEPENDENCE STRUCTURES 4.2
representation is unique, (4.2.3) is equivalent to a = o\. Since the transformation
t —> h(t) = 2c - t reflects points of R1 around c, the relation a = o\ means that
a is a symmetric measure around c. I
There exist measures a that are not symmetric around any point, e.g., a(dx) =
\{x > 0}e~xdx. If (XUX2) has the representation (4.2.6) with such a a and
some b, then the conditional distribution X2\X\ will not be symmetric. Hence,
Corollary 4.2.3 There are SaS vectors (X\,X2), 1 < a < 2 such that the
conditional laws X2\X\ are not symmetric (around the conditional mean).
On the other hand, if a is symmetric around some point c, then the distribution
of X2\X\ is symmetric around cX\, and, by Theorem 4.1.2, c will be equal to
[X2,Xl}a/\\Xl\\a.
The following corollary replaces the symmetry condition on a by a condition
on(XuX2).
Corollary 4.2.4 LetX\,X2, 1 < a < 2, be jointly SaS random variables. Then
the conditional distribution X2\X\ is symmetric (around cX\) if and only if
(x,, x2) = (y,, y2) + (y,,2cy, - y2), (4.2.7)
where Yi,Y2 are jointly SaS random variables and (Y\,Yi) is an independent
copy of (YUY2).
PROOF: To prove sufficiency, note that the vectors
(xux2) = (y1,y2) + (y1,2cy1-y2)
= (Y^+Yl,2cYi+Y2-Y2)
and
(xu2cxi-x2) = (y1 + yll2c(y1+y1)-(2cy1+y2-y2))
= (y, + y,,2cYi+y2-y2)
have the same distribution.
To prove necessity, let {Z\,Z2) and (Z\,Z2) be independent copies of
(Xi, X2). By the definition of stability and (4.2.3),
{XUX2) I 2-xla{{ZuZ2) + {ZuZ2))
± 2-^a((ZuZ2) + (Zu2cZl~Z2))
= {YuY2) + {Yu2cYx-Y2),
where (YUY2) = 2~^a{ZuZ2) and (YUY2) = 2-"a{ZuZ2). I

4.3
LINEAR DEPENDENCE
185
4.3 Linear dependence
A random variable X2 is linearly dependent on a random variable X\ if there is a
constant c and a random variable Y independent of Xx such that
X2 = cXi+Y. (4.3.1)
Jointly Gaussian random variables are always linearly dependent. As we will see,
linear dependence in the SaS case, 1 < a < 2, is related to the non-skewness
and stability of the conditional laws. (We use the term non-skewed to characterize
a law which is symmetric around some point on the real line.)
Let X\ and X2 be SaS, a < 2. If X2 is linearly dependent on X\ then all the
conditional laws X2\X\ = X\, —00 < x\ < 00, are not only symmetric (around
cx\) but also a-stable. (In Section 4.2, we did not require that the conditional laws
be stable.) The following lemma shows that in the case 1 < a < 2, non-skewness
and stability of the conditional laws hold only under linear dependence. The case
a < 1 is still open.
Lemma 4.3.1 Let X\, X2 be jointly SaS, 1 < a < 2. If all the conditional laws
ofX2 given X\ are non-skewed a-stable, then
X2 = cXi+Y
for some real c and a SaS random variable Y, independent ofX\.
PROOF: Since all the conditional laws of X2 given X\ are a-stable and non-
skewed, the conditional characteristic function of X2 given _X"i is of the form
E{^{i62X2)\Xx} = exp{-|02|QM(X,) + i62N{Xx)},
where M > 0 is a measurable function and, where, by Theorem 4.1.2,
N(Xl) = E(X2\Xl) = cXu a.s..
Let
Z = cXi + (M(X,))l/aZ0, (4-3.2)
where Z0 ~ Sa( 1,0,0) is a random variable independent of X\. Since all
conditional laws of Z given Xi coincide with the conditional laws of X2 given
X\, we have
(Z,Xl)^(X2,X1). (4.3.3)

186 DEPENDENCE STRUCTURES 4.3
This shows that Z and X\ are jointly SaS, and thus the linear combination
(M(X\)y/aZo = Z - cX\ is a SaS random variable. There is, therefore, some
a > 0 such that, for any real 9,
exp{-|0|<VQ} = EtxplieMiXxY^Zo}
= EE{exv(i6M{Xx)l'aZ0\Z0)}
= £exp{-|0rM(.X,)}-
M(Xi) = aa a.s. is a solution of this equation and it is the only solution by the
uniqueness of the Laplace transform. Using (4.3.2) and (4.3.3), we obtain
X2 = cX\ + aZ0,
proving the lemma. I
The following result shows that the SaS case, 1 < a < 2, differs greatly
from the Gaussian case.
Proposition 4.3.2 Let X\, X2 be jointly SaS, 1 < a < 2. Then both the
conditional law ofX2\X\ and the conditional law of X\\X2 are non-skewed a-
stable if and only if X\ and X2 are either independent or proportional to each
other.
PROOF: Sufficiency is obvious. Suppose that the conditional laws of X2\Xi
and X\\X2 are non-skewed and a-stable. Then, by Lemma 4.3.1, there is a SaS
random variable Y\ independent of X\ and a SaS random variable Y2 independent
of X2 such that
X2 = cXy + Y,
and
Xl±bX2 + Y2,
where c and b are constants.
Consider the first relation. Since 6\X\ + 02X2 - (0i + 62c)Xx + 02Y2, we
see, by writing down the joint characteristic function of (X\ ,X2), that the spectral
measure of (X\, X2) is concentrated on the set
of the unit circle S2 (draw a picture; refer also to Example 2.3.4). Similarly,
the relation X\ = bX2 + Y2, implies that the spectral measure of (X\,X2) is
concentrated on

4.4
PROBABILITY TAILS OF ORDER STATISTICS
187
Since the spectral measure is unique, these two sets must coincide. Clearly, this
happens if and only if the spectral measure is concentrated either on the four
poles of S2 or on two diagonally opposite points (draw pictures!). The first case
corresponds to independence (Example 2.3.5), the second to X2 = cX\. I
Remark. This proposition is clearly false in the Gaussian case because it would
read: "Any pair of Gaussian random variables has correlation zero or one." The
proof fails in the Gaussian case because the spectral measure is not unique when
a = 2 (see Section 2.4).
4.4 Probability tails of order statistics
We study in this section the probability tails of order statistics of jointly strictly a-
stable random variables X\,X2,- ■■, Xn. Because of the representation theorem
3.5.6, we can always suppose that the random variables are given by their integral
representation Xk = fBfk(x)M{dx), k — l,...,n, where M is an a-stable
random measure. The order statistics of X\, X2, ■ ■ ■, Xn are the random variables
XW > X& > ■> X^ obtained by arranging Xi,X2,...,Xn'm decreasing
order. We consider firstly X^n\ the "minimum" order statistic.
Theorem 4.4.1 Let Xk = JB fk(x)M(dx), k = 1,... ,n, where M is an a-
stable random measure with control measure m and skewness intensity (3. Then
lim XaP ( min Xk > A
A—*oo V/c—l,...,n
+ J f min n[fk(x)]t\ (1 - 0{x))m{dx)\ , (4.4.1)
where Ca is the constant defined in (1.2.9).
Notation. Here,
( x ifx>0 f 0 ifz>0
x+ = i and x- = I = (-x)+ > 0
{ 0 ifx<0 [ -x ifx<0
denote, respectively, the positive and negative part of x.
We know (see Property 1.2.15) that if Xk ~ Sa{ak,0k,O), then
lim XaP(Xk > A) = (l/2)Ca(l + &K- Theorem 4.4.1 shows that
A—* 00

188 DEPENDENCE STRUCTURES 4.4
P(min(Xi,..., Xn) > A) and P(Xk > A), k = 1,..., n, have the same
asymptotic behavior up to a constant of proportionality.
Let us consider the constant of proportionality. If Xk = JE fk(x)M(dx),
then, by Properties 3.2.2 and 1.2.15,
lim XaP{Xk > A}
A—*oo
= \ca {jE\Mx)W +0(x))m(dx) + J[fk(x)]Z(l - 0(x))m{dx)} .
Comparing this with (4.4.1), we see that lim XaP{ min Xk > A} is obtained
A—»oo fc=l,...,n
by replacing [/fc(x)]± by the pointwise minimum min [fk(x)}±.
fe=l,...,n
Remarks
1. Relation (4.4.1) can be expressed in terms of the spectral measure T of the
a-stable random vector (X\,..., Xn) as follows:
lim XaP( min Xt > A) = Ca I min [Si]ir(ds). (4.4.2)
A—»oo i=l,...,n Jc i=l,...,n
Indeed, it is easy to verify that the right-hand side of (4.4.2) and of (4.4.1)
coincide (see Exercise 4.11).
2. If M is SaS, then (4.4.1) becomes
lim XaP{ min Xk > A} ,* 4 r>\
A—»oo fc=l,...,n ^T.-r.jy
= \c° [jE {k=Tjfkix)]°+ L k^JMx)]-) m{dx)} •
3. The proof of Theorem 4.4.1 uses the series representation. The idea is
to show that the tail behavior of the first term in the series representation
dominates the tail behavior of all other terms. Using Lemma 4.4.2 below,
we conclude that only the first term contributes to the limit. The behavior
of the first term is KaX~a where Ka equals the right-hand side of (4.4.3)
Note that the series representation requires the control measure to be finite.
Our proof addresses this requirement by replacing the original control
measure m with a finite measure m, obtained from m by a judicious "rescaling."
Lemma 4.4.2 Suppose that X is a random variable with a regularly varying tail,
i.e., there is a number 9 > 0 such that for every number a > 1,
,. P(X > ax) _s
lim -^ttt r = a ■
x-oo P(X > X)

4.4
PROBABILITY TAILS OF ORDER STATISTICS
189
Suppose also that the tail of X dominates the tail of a positive random variable
Y in the sense that
lim _.,„ r = 0.
x-oo P(X > X)
Then
,. P(X + Y>x) ,. P{X-Y>x) ,
lim ———. r— = lim „.__ x— = 1.
x-oo P(X > x) x->oo P(X > X)
PROOF: For x > 0 and any a, 0 < a < 1, we have
P(X + Y > x)
= P{X + Y>x,Y>ax) + P{X + Y>x,Y<ax)
< P(Y > ax) + P(X > (1 - a)x)
so that
P(X + Y>x) _. P(Y > ax) + P(X > (\ - a)x)
hmsup „.,. — < hmsup — „,„■-—r^ —L
x-ooH P(X>x) ~ X^J P{X>x)
PlY > ax) P{X > ax) ,. P(X > (1 - a)x)
* hZ™P P(X > ax) P(X > x) + hf^P P(X > x)
= (1-*)-'•
Since a is arbitrary and since, obviously,
,. . cP{X + Y>x) ^,
hmmf—' ,. r— > 1,
x-oo P(X >x) ~
the first part of the lemma follows. The second part can be proved in a similar
way. I
Proof of Theorem 4.4.1: Let f*{x) = max |/k(a;)|, x e E. Clearly /* e
fc=l,...,n
La(m). Let m(dx) = f*(x)am(dx). Then m is a finite measure on (E,£) and
we denote by M an a-stable random measure on (E, £) with control measure fh
and the same skewness intensity/? as M. Iffffc(x) = fk(x)/f*(x), fc = l,...,n,'
then
(XuX2,...,Xn)±(YuY2,...,Yn), (4.4.4)
where
Yk= gk(x)M{dx) + Hk, k-l,...,n,
Je
'We assume without loss of generality that f'(x) ^ 0 for all i€E. Alternatively, one may set
gk (x) = 0 when /* (i) = 0.

190 DEPENDENCE STRUCTURES 4.4
and
[0 ifa£ 1,
I lSBfk(x)0(x)ln\r(x)\m(dx) ifa=l.
Note that |<7jt(x)| < 1 for all k = 1,... ,n and x € E. The YkS have series
representation (see Theorem 3.10.1):
oo
Zk = (Cam(E)Y/a Y^fai^aKW ~ bi(a>9k)), k = 1,... ,n,
where the V^s are i.i.d. with distribution tuq = ffi/fh(E) and the bi(a, gk) are
constants. We will prove
lim XaP{ min Zk > A}
A—»oo fc=l,...,n
= \c<*(f h+(x)a(l +P(x))m(dx) + J h-(x)a(l-0(x))m(dx))
(4.4.5)
where
h+(x)= min [A(x)]+, h-{x) - min [/fc(x)]_. (4.4.6)
fc=l,...,n «=I,...,n
By (4.4.4), this will imply (4.4.1).
For k = I,... ,n, set Zk = Uk + Wk, where
Uk = (^mi^l^hr-'^t^) - 6,(a>fffc)],
oo
Wt = (Cafh(E))i/a Y,frT71/a9k(Vi) - bi(a,gk)].
-l/a
Since
min C/fc - max |V^fc| < min Zk < min Uk+ max |Wfc|,
fc=l,...,n fc=l,...,n fc=l,...,n fc=l,...,n fc=l,...,n
Relation (4.4.5) will follow (see Lemma 4.4.2) if we prove
lim XaP{ min Uk > X}
A—>oo fe=l,...,n
-Ca(J h+(x)a(l + p(x))m(dx) + J h-(x)a(l - P(x))m(dx))
1
2l
(4.4.7)
and
lim XaP{ max \Wk\ > X} = 0. (4.4.8)
A—>oo fc=l,...,n

4.4
PROBABILITY TAILS OF ORDER STATISTICS
191
We first establish (4.4.7). We may obviously assume b\(a,gk) = 0. For A > 0,
P{ min Uk > A}
fc=l,...,n
- p{(cani(g))1/arrl/amin*"'/;^A(Vi) > a}
= JsP ({CamW)i'*T7l/a^ > A) ^|Mm0(dx)
+ /eP ((Cam(£0)^IT,/o^ > A) ^mo(dx)
-/(,-^^(^)'})l^W
+ /£(l-eXp{-C^)A^(^)Q})iz|(£)mo(&).
(4.4.9)
Now (4.4.7) follows from (4.4.9) and the dominated convergence theorem.
n
We now prove (4.4.8). Since P{ max \Wk\ > A} < V* P{\Wk\ > A}, it
fc=l,...,n , ^—'
fc=l
is enough to show (see Exercise 4.14) that E|Wfc|Q < oo for all k = 1,... ,n.
Let Wk be an independent copy of Wk given by
oo
Wk = (CQm(E))'/Q £[7ifr,/a5fcM) " hi(a.Sfc)].
»=2
where {(7,,^,^), i = 1,2,...} is an independent copy of {(7,, r^Vj), t =
1,2,...}. Let
00
Rk = (Cain(£))-"Q(ft - Wk) = ^(7irr1/Q5fc(Vi) - 7if7,/a<fc(V0).
It is clearly enough to prove that E\Rk\a < 00. Write Rk = Rkl) + Rk2), where
OO
00
42) = E(rr!/Q-Fr1/Q)7i5fe(v;).
i=2
Then we need to show that E\Rkj) \Q < 00 for j = 1,2. Consider first j = 1. Let
{ei, i> 1} be a sequence of i.i.d. Rademacher random variables independent of

192
DEPENDENCE STRUCTURES
4.4
all other random variables in Rj*' and let io = [2/a] + 1. Then
^(E^fc(Vi)-7iM(Vi))rr,/01>)
= E (f; e4(7iS*(Vi) - 7iflfc(^i))rr1/a )
OO
i=io
oo oo
< 4 £ ET~2/a <4CJ2 i~Va < oo
using Exercise 4.15 and |<7fc(x)| < 1- Moreover, for any t = 2,..., io — 1,
E (|7i9fc(Vi) - 7i5fe(V^)|rr1/a)Q < 2a£Tr! < oo.
Hence ^l-R^V is bounded by
to —1 oo
*o [E ^l(7iflfe(Vi)-7i9*(V5))rr,/a|a+£| E(7i3fe(^i)-7i5fc(^))rr1/Qr]
(2) I
which is finite. It now remains to prove that E\Rk \a < oo. We use Exercise 4.16
to conclude that there is a j\ such that for any j > j\, Eexp{j~l^2\Tj —j\} < 4.
Choose, now, a number 9 > 2 such that g + £ > 1. By Markov's inequality, for
any A > 0,
P{\T:-j\ > Aj*-1/e) = PtexpO-'/2^- - j\} > expW2-1/*})
<4exp{-Aj'/2-e}.
Therefore, for some constant b\ — b\ (9) and any A > 1,
oo oo
£ P(\rj-j\>\jl-l/B) < 4 52 exp{-Aj'/2-/e}
< 4 Pexpi-Ax1/2-1/"}^
< 6, A~4 (4.4.10)
since A4e~Al , e > 0, is integrable. By the mean value theorem,

4.4
PROBABILITY TAILS OF ORDER STATISTICS
193
and hence
p(|r71/a-f7,/a|>Ar(i+i))
< PQTj ~fj\> Aamifl(rj,rj)I+,/ar(*+i))
< PivamiTjfj) < jA~^hT) (4.4.11)
+p(|rj-f,|>aA1/V-n
Set a = q/2(q + 1). For any A > 0 large enough, by Stirling's formula,
P(min(ri,fj)<jA-Q) < 2P(rj<jA-a)
- 2l irvr»
where 62 is a finite constant. Therefore
00 00
£ P(min(rj,fj)<jA-a) < 62 X! J'3/2(eA~°)J
< 63(eA-a)j,+1 *
1 - eA~°
< fc4A"0,+l)a
< M-2
by choosing A and ji large enough. Hereby i = 3,4, are finite constants. Further,
by the triangle inequality and (4.4.10), for any A large enough,
oo
J2 P(\Tj-fj\>aX^j^e)
CO
<2 £ P(\Tj-j\>^j^)<b5X-'
for some finite constant 65. Therefore, for large enough ji, we obtain
.r-.l/a_j;-l/o| v
CO
< Y, iD(|r71/a-f71/a>Ara+°))<0(A-2), (4.4.12)

194 DEPENDENCE STRUCTURES 4.4
and, for such j\,
oo
' p(\ E (r71/Q-?7I/a)^fc(^)l>A)
oo
<^(E |i7,/"-fj-"-|>A)
ip-l/a _ p-'/<*| oo
<f(sup' ' , ' ' E r(*+±> > A) < 0(A"2)
because g + £ > 1. Since, for every j > 2,
E\(rj'/a -f7I/a)7ifffc(vi)|a < £|r-1/a _f;1/Qr < 21+tt£;r71 < oo,
we conclude that E\Wk '\a < oo. This proves the theorem. I
Corollary 4.4.3 Under the conditions of Theorem 4.4.1,
lim XaP ( min \Xk\ > A J = Ca I min \fk(x)\am{dx).
A-»oo \&=l,...,n J Jg fc=l,...,n
PROOF: Let 5 = {(si,..., sn) : sk = ±1, k = 1,..., n} be the set of all
n-tuple possible "signs." Since
P{ min |A-fc| > A) = P(|X,| > A,..., \Xn\ > A)
fc=l,...,n
E P(s\X\ >\,...,snXn>\),
(si,...,«„)65
lim AQ( min |Xfc| > A) equals the right-hand side of (4.4.1) with
A—>oo k=l,...,n
min [/k(x)]± replaced by £, s )e5minfe=i,...in[3fc/fc(a:)]±. Fix now
fc=l,...,n v ' "' "'
x € E and observe that the minimum is zero if /k(z) = 0 for some
k. Suppose /fc(x) ^ 0 for all fcs. Then minfc[sfc/fc(x)]+ ± 0 only if
Sfc = sign (/fc) for all fcs, in which case minfc[sfc/fc(a;)]+ = mint |/fc(a;)|.
Similarly, minfc[sfc/fc(z)]_ ^ 0 only if sjt = — sign (fk) for all ks in which case we
obtain again minjt[sfc//t(x)]_ = mint |/fc(x)|. This proves the corollary. I
Theorem 4.4.1 gives the tail behavior of the minimum. In order to obtain the
tail behavior of any other order statistic, we use an inclusion-exclusion type
formula (see Lemma 4.4.4 below) which relates any order statistics to the minimum.

4.4
PROBABILITY TAILS OF ORDER STATISTICS
195
Lemma 4.4.4 Leta\, a2, ■.., an be real numbers and let a^ > a^ > ■■■ > a^
be a non-increasing permutation of these numbers. Then
«(fc) =£(-!)''"* ({I1, ) £ min(ai|,ai2)...)aij).
j—k ^ ' l<ii<t2<-<ij<n
Proof: See Exercise 4.18.
Theorem 4.4.5 Let Xi,Xi,...,Xn be jointly strictly a-stable random variables,
given in the form Xk = jEfk(x)M(dx), k = l,...,n, where M is an en-
stable random measure with control measure m and skewness intensity (3. Let
XW > > X^ beXuX2,...,Xn rearranged in nonincreasing order. Then,
for anyk = 1,2,... ,n
lim AQP(X<fc> > A)
A—+oo
- \ca U hkt+(x)a(l + (3{x))m{dx) + J /ifc,_(x)a(l - f3(x))m(dx)\ ,
(4.4.13)
where
hk,+ (x) = fcth largest among [fi{x)]+, i = 1,..., n,
hk,-{x) — fclh largest among [/i(x)]_, i = 1,... ,n. (4.4.14)
PROOF: Set ak = I(Xk > A) in Lemma 4.4.4 to obtain
lim XaP(Xw > A)
A—»-oo
= D-^fill) E JBmA-P(^I>A,...,^>A)
j=k x ' l<ti<...<tj<n
D-'^-fi:',) E 5c«
j=fc v l<ii<...<tj<n
X
(/ min I/id(i)]^(l+/3(x))m(di)
+ / min [/id(s)£(l-/3(*))m(da:)
= icQ | J hkt+(x)a(l + P(x))m(dx) + J hk,-(x)a(l - P(x))m(dx)} .
We used Theorem 4.4.1 in the second step, and in the last step, we used Lemma
4.4.4 once more. I
Remark. The tail of the distribution of any order statistics behaves like A~a as
A —► oo.

196 DEPENDENCE STRUCTURES 4.4
Corollary 4.4.6 Let \X\^ > ■■■ > \X\^ be a non-increasing permutation of
\X\ |,..., \Xn\. Under the conditions of Theorem 4.4.5,
Urn A°P(|X|(fc) > A) = Ca f hk{x)Qm(dx),
A-.00 JE
where hk{x) is the felh largest among \fi(x)\, i = 1,..., n.
The proof is similar to that of Corollary 4.4.3 and is left as an exercise.
As an application of the preceding results, we will illustrate the difference in
behavior of linu_oo P(X2 > A \X\ > A) between the Gaussian and stable cases.
Example 4.4.7 The conditional probability P(X2 > A |Xi > A), A > 0, is a
measure of dependence. If Xi large implies that X2 is likely to be large, then X\
and X2 are "positively" dependent. If it implies that X2 is unlikely to be large,
then X\ and X2 are "negatively" dependent.
Suppose, firstly, that (Xi, X2) is a mean zero Gaussian vector. Then X2 =
cX\ + Y where Y ~ N(Q, o2) is independent of X\, and (see Exercise 4.20)
!1 ifc> lorifc= 1,ct = 0,
0 ifc< 1, (4.4.15)
1/2 ifc=l,cr>0.
In the stable case, on the other hand, lim P(X2 > A \X\ > A) can be any
A—too
number in [0,1]. Let Xk = JE fk{x)M(dx), k = 1,2, where M is a SaS random
measure with control measure m. By Theorem 4.4.1,
hm P{X2 > A X, > A) = hm ■
A—>oo A-*oo P\-X-\ > A)
^ JBmin((/i(g))+>(/2(»))+)am(d») + Jgmin((/,(g))-,(/2(8))-)am(da)
/E(/i (x))%m{dx) + JB</, (*))« m(di)
^ Jg, |/.(x)l°m(dx) + Jg; |/2(a)|°m(ds)
JE\f\(x)\am{dx)
where
JS, = {xeE:f2(x)^0, 0<h^\<\\,
<• 72 (x) J
Ek = {xeE:f2(x)^0,^>l}.
(Note that f\(x) and /z(x) have the same signs on E{ and E2.) By choosing /i
and f2 suitably, lim P(X2 > A |Xi > A) may take any value in [0,1].
A—>oo
(4.4.16)

4.4 PROBABILITY TAILS OF ORDER STATISTICS 197
The statement of Corollary 4.4.6 with k = 1 can be rewritten in the form
^ AQP(||X|| > A) = CarH{S*\ (4.4.17)
where || ■ || is the L°°-norm on Rd (use Theorem 3.5.6, Part (i)). It turns out
that (4.4.17) remains true for any norm on Rd, and, in fact, the statement can be
strengthened.
Theorem 4.4.8 Let || ■ \\ be a norm on Rd, and letX= (Xu..., Xd) be an a-
stable random vector whose characteristic function is given in Proposition 2.3.8.
Then for every Borel set A C SJj'11 with T\\.\\ (dA) = 0,
^ AQp(||X|| > A, pj| 6 A) = CaTH(A). (4.4.18)
PROOF: The proof is similar to that of Theorem 4.4.1. We first use Theorem
3.5.6 and Theorem 3.10.1 to conclude that (XUX2,.. ■ ,Xd) = (ZUZ2,. ..,Zd)
where
oo
Zk = {CaTM^))'laY^:X,a^k)-Kk{a)l k= 1,2,...,d.
i=\
Here S* = (Sj1 \... ,s\d ]), i = 1,2,..., are i.i.d. sj"11-valued random
vectors with common law r||.||/r||.y(S)J ) and h^ia), i > 1, k = 1,... ,d, are
constants. If
uk = (car|HI(s|S-|l)),/a[rr,/as{fc)-6l,fc(a)],fc=i,...,d,
wk = (carll.|l(s»-|')),/0f;[rr1/a5|fc)-bi,fc(a)],fc = i,...,d
andU = (l7i,...,[/d),W = (W,,...,Wd),thenZ = U + W.
For any 0< e < 1/2,
p(\\z\\>x,^eA)
< P(||U|| >(1 - 6)A, ||W|| < eA, + eA)+ P(||W|| > e\).
(4.4.19)
Using the equivalence of norms in Rd and (4.4.8) we conclude that for some finite
positive constant c,
limA-.OQAQP(||W|| > eA)
< IiHA_00AaP(c||W||00 > eA)
= cQe-Q lim AQP( max |Wfc|>A)=0. (4.4.20)
A—oo Vfc=l,...,d

198
DEPENDENCE STRUCTURES
4.4
Suppose, now, that the set A is closed, and for cr > 0, let Aa denote the closed
CT-neighborhood of A. Since
u + w u
<
2||W||
l|U + W|| ||U|| -||U + WU'
we conclude that
p(||U|| > (1 - e)A, ||W|| < eA, ]jH±W_ ei,^ A^ = 0,
where e' = 2e/(l - 2e). Therefore
/>(||U||>(l-e)A1||W||<eAf]iH±W.€A)
<p(||U||>(l-e)A,~€^). (4.4.21)
SettingU<0) = (Ul0),...,1/W) with
Ma
^0) = (^r,l|(4«)) rT1/as[k\k = i,...,d,
and b* = maxfc=i?...id |6iijfe(o;)|, we can choose A large enough so that
U U<°>
||U-U<°>||<Ae and
||U|| ||U(«»||
<e'.
For such A, if e" = 2e' = 4e/( 1 - 2e), the right-hand side of (4.4.21) is bounded
above by
p(||U<0>||>(l-2e)A,
U(»)
pjwjj
eA.
<■)
= p(rr"° > (i-2<)(c„r,.||(sJll))-"<'A,s, e v)
= p(r,<(1-2e)-»c„r,,(sr')A-»)M^.
1 11-11 Wd /
Therefore, for any 0 < e < 1,
JMX-.00\ap(\\Z\\ > \,~£- 6 A)
< limA-»00AQp(r1 < (1 -2£)-aCQr|M|(S|{il)A-Q)-
= (i-2C)-°car||.ll(v)-
■!(#■)
(4.4.22)

4.4 PROBABILITY TAILS OF ORDER STATISTICS 199
Letting t \ 0 and noting that Ae» | A as 11 0, we conclude from (4.4.22) that for
any closed A,
IiSA^00Attp(||Z|| > A, p| € A) < CQrH(A). (4.4.23)
On the other hand, we have P(||U|| - ||W|| > A) < P(||Z|| > A) < P(||U|| +
||W||> A), and hence
^A^PdlZH > A) = C^ry.,,^11) (4.4.24)
by Lemma 4.4.2. Clearly, (4.4.23) and (4.4.24) imply that as A —> oo, the measure
Px defined by PA(A) = AaP(||Z|| > A,Z/||Z|| € A) converges vaguely to
CQr||.||. Therefore (4.4.18) holds. I
The following is an extension to Property 1.2.18.
Corollary 4.4.9 Let || ■ \\ be a norm on Ed, and let X = (Jt,,..., Xd) be an
a-stable random vector. For 0 < r < a, let T^l be the following finite Borel
measure on SJj'":
Tl;\ = (a-r){aCa)-lPg>oT-\
where P§[\dx) = ||x||rPx(dx), Px is the law o/X, and T: Rd\{0} -* SJj'" is
given by Tx = A. Then asr 1 a, lYl converges to ry.y vaguely, i.e., for every
continuous function h on SJJ'", we have
f h(s)r\{\(ds)-^ /" h(s)r,|.B((fa) asrTa.
PROOF: Since the measures ri^j!, 0 < r < a, and Ty. y are concentrated on Sd ,
we only need to show that for every Borel set A C SJj" with r||.||(9A) = 0, we
have
limrji:j(A)=r||.ll(A). (4.4.25)
But for any 0 < r < a and any Borel set ACSj ,
T\;\(A) = (a-r)(aCa)-1 f ||x||rPx(dx)
11'" J{xeRd\{0},x/||x||6^}
= (a - r)(aCQ)-'J5||X||r 1(X/||X|| € A)
JrOO
' p(||X||>tl/p,X/||X||€4)(ft.
o
The proof concludes by following the argument in the proof of Property 1.2.18
and then using Theorem 4.4.8. I

200 DEPENDENCE STRUCTURES 4.5
4.5 Joint moments
Let X = (Xi,..., Xn) be an a-stable random vector in Rn, 0 < a < 2, and let
Pi,..., pn be non-negative numbers. We want to find a necessary and sufficient
condition on p\,... ,pn for the joint moment £|Xi|Pl • • • |Xn|p" to be finite.
We suppose throughout that X is non-degenerate, i.e., that its distribution is not
concentrated on a point.
If the X,- are independent, then £|X, |Pl •••|X.|P" = £|X,|Pl ■ ■ ■ E\Xn\p~ <
oo if and only if pj < a, j = 1,... ,n. If the Xj are all equal (say Xj =
X,, j = 2,... ,n), then £|X,|P' ■ ■ ■ |X„|P« = £|X,|p'+"+p" < oo if and only
if Pi H hp„ < a. We use the integral representation Xj — JE fj(x)M{dx) +
rjj, j — 1,..., n, in (3.5.11) (see Theorem 3.5.6 (i)) to obtain a necessary and
sufficient condition for £|Xi|Pl • • • |^n|p" < oo in the general case. Suppose
firstly that the XjS are n-fold dependent, i.e.,
m{xeE: fi(x)f2(x) ■ ■ ■ fn{x) ± 0} > 0.
When n = 2, n-fold dependence is equivalent to dependence by Theorem 3.5.3.
When n > 2, the random variables can be dependent without being necessarily
n-fold dependent.
Example 4.5.1 Let 17, V, W be three disjoint subsets of E with positive m
measure, and let Xj — jEfj{x)M{dx),j = 1,2,3, where /i,/2, /shave support C/U
V, V U W, W U U, respectively. The random variables X\, X2, X3 are dependent,
in fact pairwise dependent, because m{x e E : fi(x)fj(x) ^ 0} > 0, i ^ j, but
they are not 3-fold dependent because m{x € E : fi(x)f2{x)f3(x) ^ 0} = 0.
The spectral measure T can also be used to express the dependence structure
of (X\,... ,Xn). The XjS are independent if and only if T is concentrated on the
2n points (0,... ,0,±1,0,... ,0) of the unit sphere Sn of Rn. They are n-fold
dependent if and only if
r{seSn:si^0,S2^0,...lSn^0}>0.
Lemma 4.5.2 Let X\,..., Xn be jointly a-stable and n-fold dependent. Then
E\XX |Pl • • • |X„|P» < 00 if and only i/pi + • • • + p„ < a.
PROOF: Necessity. Suppose E|Xi|Pl • • ■ |Xn|p" < 00 and let A = {x € E :
f\(x) ■ ■ ■ fn{x) ? 0}. For each j = 1,...,n, define Yj = JA fj(x)M(dx)
and Zj = JAcfj(x)M(dx). The vectors {Yu...,Yn) and (Zu...,Zn) are
independent because A and Ac are disjoint. Therefore the hypothesis
00 > £|X!|PI • • • |Xn|p" = E\Yi + Zx + r/,|PI • • • \Yn + Zn+ 7?n|p"

4.5
JOINT MOMENTS
201
implies there are constants Z\,..., zn such that
co > E\Yl+zl\^--\Yn + zn\^
= / ■••/ PQYi+ztl* >xu...,\Yn + zn\P" >xn)dxl...dxn
Jo Jo
> -/(A)
for any A > 0 where J(A) equals
/•A" /-A'"
/ •••/ P(\Y + Zl\p> >xu...,\Yn + zn\P">xn)dxi...dxn.
(4.5.1)
Since J(A) is the tail of a convergent integral, we have
lim J(A) = 0.
A—>oo
On the other hand, replacing each Xj in (4.5.1) by the upper bound Ap^, j =
1,..., n, yields
J(A) > CAp'+"+p"P(|Y1+z1|>A,...,|yn + zn|>A)
> CX»+-+P«P{\YX\ > 2A,..., \Yn\ > 2A)
for some positive constant C and for all large enough A. By
Corollary 4.4.3, there is a positive constant C\ < C, such that J(A) >
C,2-QCaAPl+-+p"-a /,( min \fk{x)\a)m{dx) for all large enough A. The
fc=!,...,n
integral on the right-hand side is positive because n-fold dependence implies
m(A) > 0. Since J(A) —> 0, we must have p\ -\ (- pn < a.
Sufficiency. If pi 4- \-pn < a, then, by the Holder inequality,
n
E\Xl\p'---\Xn\Pn < Y[{E\Xk\p'+-+p")p^^+-+p^ < co.
fe=i
This completes the proof. I
Remark. The assumption of n-fold dependence is not used to prove sufficiency.
Example 4.5.3 If X\ and X2 are jointly a-stable, q < 2, and not independent,
then they are 2-fold dependent. Consequently, E\XX |Pl l-X^I?2 < co if and only if
Pi+Pi<a.
We introduce now the notion of maximal subset in order to analyze the general
case. Let {k\,..., km} denote a subset of size m of {1,2,..., n} and let
Ak fcm = {x€E: fkl{x)... /fcm(x) * 0} (4.5.2)
be the intersection of the supports of /*,,..., fkm ■

202 DEPENDENCE STRUCTURES 4.5
Definition 4.5.4 The subset {ki,...,km} is maximal if rn{Ak,,...,km} > 0 and
m{Aku...,km,i} = 0 for any i <? {A;,,..., km}.
Example4.5.5 If Xi,X2,X3 are as in Example 4.5.1, then {1,2}, {2,3} and
{3,1} are the three maximal subsets of {1,2,3}.
Theorem 4.5.6 Let X = (X{,... ,Xn) be a non-degenerate a-stable random
vector. Then .E|X||Pl • • • |Xn|Pn < oo if and only ifpk, + ■ • • + Pkm < a for
every maximal subset {k\,... ,km} of {1,2,... ,n}.
Proof: We suppose without loss of generality that
(X,,...,Xn) = (J h{x)M{dx),...,J fn(x)M(dx)Y
Necessity. Suppose .E|Xi|Pl • ■ • |X„|Pn < oo and let {ku---,km} be any
maximal subset of {1,2,..., n}. We want to show that pk, +•■•.+ pkm < ot.
Define
Yj = I fj{x)M{dx) and Z, = f fj(x)M(dx),
■>Ak km JA'k fcm
j = 1,..., n, where Aku...,km is given in (4.5.2). Since {k\,..., km} is maximal,
for every j g {k\,..., km} we have/^(x) = Om-almost everywhere on Ak,,...,km
and, therefore, Yj = 0 a.s. By hypothesis,
oo > £|X,|p'---|Xn|p«
= E\Yk,+Zk,\^---\Ykm + Zkn\"^ 11 \Zj\n.
j£{ku...,km}
The independence of (Y\,...,Yn) and (Zu...,Zn) implies E\Yk, +
Zfc,lpt| " • • \Ykm + ZkJPkm < oo. Since m(Ak km) > 0, the vector
(Yjt,,..., Yj;m) is m-fold dependent and therefore, by Lemma 4.5.2, we have
Pk, +---+Pkm <a.
Sufficiency. For every subset {fci,..., km} of {1,..., n} let
B({ki,...,km})
= {xeE:fkt(x)-~ fkm (x) ? 0, fi(x) = 0, Vi 0 {fc,,..., km}}.
Since the sets {B({ku... ,km}) : {ku...,km} e 2*1 ">, 0 < m < n}1
partition Ujt=i SUPP(A)> we can write
2{fel,..., km} is expressed here as an element of 2t'-—•"}, the set of all subsets of {1,..., n}.

4.5
JOINT MOMENTS
203
Xj = Yl Xj({ku...,km}), j = l....,Ti,
{fcl fcm}
where
Xj({fci,...,km}) = / fj(x)M(dx)
JB({fci fcm})
and E' means that the summation is over all subsets {k\,... ,km} satisfying
m{B{{ki,..., km})} > 0. (To simplify the notation, we shall write in the sequel
B(k\,..., km) and X{k\,..., km).) Since, for some constant Ca,n<
E\X,r ■ ■ ■ \Xn\p" < Ca,nB f[ £ l^(fc[r), • • •,*&>)!*,
r=1{k[r) *&>.}
it is enough to prove
E\Xl(k\1\...,k^)r-..\Xn(k\n\...,k^n)\^<cX> (4.5.3)
for all subsets {fc[r),..., k£l} of {1,..., n} satisfying m(B(k\r),..., fcj#)) >
0, r = l,...,n. Some of the subsets {k\ , ...,fcm'}, r = l,...,n, may
coincide. Let K denote the collection of all different such subsets. Observe
that the cardinality of K is at most n. Now, for each {k\,..., km} € K, let
I(k\,...,km) denote the set of all rs as above such that {fc, ,...,kml} —
{fc,,..., km}. Since X^kf*,..., fc^) and X,(fcjj),..., k&]) are independent,
II {fc[ , . . . , fcmi j 7= {fcj , ...,fcmj/i
E|x1(fc|i),...,fc^)r---ixn(fc|n),...)^)ip"
= n ^ n i^(*i.--.*m)iw-
{fel,...,fcm}eJ<: je/(fci fcm)
It is therefore enough to prove that
E [J |Xj(fc,,...,fcm)|w<oo (4.5.4)
je/(fe|,...,fcm)
for any {k{,...,km} satisfying m(B{k\,... ,km)) > 0. Clearly, if j i
{ku...,km}, then Xj(ku... ,km) = 0, so we may assume in (4.5.4) that
j G {fci,...,fcm} for every j e I{ki,... ,km). In that case, the random
variables {X,-(fci ,...,km), j € I(kt,..., fcm)} are m-fold dependent. Since
{k\,..., km} is a subset of some maximal subset {k\,..., fcm, fcm+i,..., km<}
of {l,...,n}, we have
X! Pj<Pfc,+---' + Pfcm+Pfcm+1+---+Pfcm' <Q
ie/(fci,...,fcm)

204 DEPENDENCE STRUCTURES 4.6
by the sufficiency hypothesis. Relation (4.5.4) therefore follows from Lemma
4.5.2. ■
Example 4.5.7 If X\, Xz, Xj are as in Example 4.5.1, the three maximal subsets
are {1,2}, {2,3} and {3,1} and therefore E\Xi |Pl |X2|K|X3|P3 < oo if and only
if P\ + P2 < a, P2 + Pi < a and P3+P1 < a.
Remark. No inequality in the system
[pfc, + H p*m < a, {fci,..., fcm}
is a maximal subset of {1,2,... ,n}, 1 <m <n\
is redundant in the sense that no inequality is implied by the others. To verify this,
consider a particular inequality, say Pk, + ••• + Pkm < a. The choice of values
Pk, = • ■ • = Pkm = ~, Pj = 0 for j ^ ku...,km violates this inequality, but
it violates no other in the system because {fci,..., km} is a maximal subset and
hence no other inequality in the system can include more than m — 1 of the pis,
l = K\, K2, ■ ■ ■ , km.
4.6 Association of stable random variables
Random variables X\,..., Xn are said to be associated if, for any functions
/, g : Rn —> R which are non-decreasing in each argument, one has
Cov{f(Xi,...,Xn),g{Xu...tXn))>0,
whenever the covariance exists. The following theorem gives necessary and
sufficient conditions for a-stable random variables with 0 < a < 2 to be associated.
Theorem 4.6.1 Let X\,..., Xn be jointly a-stable random variables, 0 < a < 2,
with spectral measure T on the unit sphere Sn of Rn. Then X\,... ,Xn are
associated if and only if the spectral measure T satisfies the condition
r(S-) = 0, (4.6.1)
where
S~ = j(si,...,sn) s Sn : for some i,j € {l,...,n}, Si > 0 and Sj <0j.

4.6
ASSOCIATION OF STABLE RANDOM VARIABLES
205
PROOF: Necessity. Suppose to the contrary that the X\,..., Xn are associated
and T(S~) > 0. There then exist i, j € {1,..., n) such that
r{(si,. ■ ■, a„) € Sn : Si > 0, 3j < 0} > 0. (4.6.2)
Without loss of generality we assume that i = 1, j = 2. Proposition 2.3.9 together
with (4.6.2) imply that Xi, X2 are jointly a-stable in E2 and the spectral measure
I\2 of (X\, X2) satisfies
ri,2{(si,s2) G S2 : s, > 0, s2 < 0} > 0. (4.6.3)
Moreover X\, X2 are associated, being a subset of a set of associated random
variables {X\,... ,Xn}. For any A > 0, the association of X{ and X2 implies
(see Exercise 4.23)
P{Xi > A, X2 > -A) > P{XX > X)P(X2 > -A)
and hence
P(X2 < -A \Xi > A) < P{X2 < -A). (4.6.4)
Clearly, the right-hand side of (4.6.4) goes to zero as A goes to infinity. Moreover,
as X[ and — X2 are also jointly a-stable, with spectral measure T\t2 defined by
F,i2(i4) = r,,2{(s1)s2) 6 52 : (s,, -s2) e A},
we conclude from (4.4.2) and (4.6.3) that
lim P(X2 < -A |Xi > A)
A—* 00
= lim P{-X2 > A |Xi > A)
A—*cx)
= JS2(min([3i]+,H+))ar1,2(ds)
/a([s,]+)arIi2(dS)
4([Sl]+)°r,,2(dS)
The contradiction thus obtained proves the necessity part of the theorem.
Sufficiency. Suppose that (4.6.1) holds. Then the spectral measure T is
concentrated on S£°s U 5nes, where
Sr = {(» «n)6 5n: Si>0, i=l,....n},
Snnes = {(s,,...,sn)e5„: Si<0, *=l,...,n}.
We may and will assume until the end of the proof that the shift vector /x° in the
characteristic functions of (X\,..., Xn) and all other a-stable random vectors

206
DEPENDENCE STRUCTURES
4.6
defined later is zero, since subtracting the shift does not affect the association
property.
Let X+ = (X,+ ,..., X+) and X~ = (Xf,..., X~) be two independent
Q-stable random vectors in Kn with spectral measures r+ and T~, respectively
given by
T+(A) = T(A n SP°S), T~(A) = T((-A) n S™s)
for any Borel set A on Sn. Clearly,
(X, ,..., Xn ) — (Xi ,..., Xn ) = (X\,..., Xn),
so that it is enough to prove both that Xf,..., X+ are associated and X,~,..., X~
are associated. Since both r+ and F~ are concentrated on Sn°s, we assume, to
simplify the notation, that the spectral measure T of the a-stable vector (X\,..., Xn)
is concentrated on Sn° » and prove that the random variables X\,..., Xn are
associated.
Let M be an Q-stable random measure on Sfi°s with control measure T and
skewness intensity 1. Then (X\,...,Xn) — (YJ,..., Y„), where
Yi= SiM(ds), i=l,...,n.
Jsr
By the definition of the integral, for each i = 1,..., n, there is a sequence of
simple Borel functions from S%°s to R, denoted {fm \ m = 1,2,...}, such that
Y$ tends to Y{ in probability as m -» co where Ym = /si»» /m)(s)M(ds),
m = 1,2,... . Clearly, we may always choose the functions fm such that
fm (s) > 0 for every m, i, and s. Suppose that
/^)(s)=E/im,i)l(sG^m'i))1i=l,---,nIm=lI2,...,
for some/jm,i) > 0, i=l,...,n, m=l,2,..., j = 1,... ,Km^, where
AJ"1'0,..., -A^] is a partition of 5^s into Borel sets. Since the measure M is
independently scattered, for each fixed m = 1,2,..., the collection
cm = {M{A^'l) n ^'2) n... n A^n)),
where ji = 1,..., Kmy,...; jn - 1, • • •, Km<n}
is a collection of independent, and therefore associated, random variables. Since

4.6 ASSOCIATION OF STABLE RANDOM VARIABLES 207
we conclude that each Y4 is a non-decreasing function of random variables from
Cm- Therefore Ym1',..., Ym are associated for any m — 1,2,... (see Exercise
4.25). But, as m -► oo, (y£} ,..., y£°) tends to (Y,,..., Yn) in probability, and
association is preservedunder convergence in distribution. Therefore, Y\,..., Yn
are associated, which completes the proof of the theorem. I
As a by-product of the proof of Theorem 4.6.1 we obtain an equivalence
of several different types of positive dependence for jointly a-stable random
variables, 0 < a < 2. Note that in the Gaussian case such equivalence follows
directly from Pitt (1982).
Following the terminology of Brindley and Thompson (1972), we call random
variables X],..., Xn F-positive orthant dependent (FPOD) if
P{XX >xx,...,Xn>xn)>P{Xx >xx)---P(Xn>xn)
for any xx,... ,xn, and we call them F-positive orthant dependent (FPOD) if
P{XX <xx,...,Xn<xn)>P{Xx < Xl) ■ ■ ■ P(Xn < xn)
for any x\,...,xn- Clearly, association implies FPOD and FPOD and, in general,
these implications cannot be reversed (see Exercise 4.26).
Corollary 4.6.2 Let Xx,..., Xn be jointly a-stable. Then all the notions of
positive dependence above are equivalent, and each of them is equivalent to
(4.6.1).
PROOF: By Theorem 4.6.1, it is enough to prove that each of FPOD and FPOD
implies (4.6.1). The fact that FPOD implies (4.6.1) follows directly from the proof
of the necessity part of Theorem 4.6.1 (in that proof, we did not use the full force
of the association property, but only its consequence, the FPOD property.) The
same argument applied to the vector {-X\,..., -Xn) shows that FPOD implies
(4.6.1) as well. I
Finally, we give necessary and sufficient conditions for the negative
association of jointly a-stable random variables. We cannot simply reverse the
definition of association because no non-degenerate random vector X can
satisfy Cov(/(X), g(X)) < 0 for all non-decreasing function / and g (to see this,
take / = g). Following Alam and Saxena (1982), we call random variables
Xx,..., Xn negatively associated if, for any 1 < k < n, any / : Rk -* R,
g : Rn-fc —* R, non-decreasing in each argument,
Cov(/(Y),S(Z))<0

208 DEPENDENCE STRUCTURES 4.7
whenever the covariance exists, where Y and Z are any k- and (n - /c)-dimensional
random vectors, representing a partition of the set (Xi,..., Xn) into two subsets
of sizes k and n - k, respectively.
Theorem 4.6.3 Let X\,..., Xn be jointly a-stable random variables, 0 < a < 2,
with spectral measure T on the unit sphere Sn of Kn. Then Xu...,Xn are
negatively associated if and only if the spectral measure T satisfies the condition
HS+) = 0, (4.6.5)
where 5+ = {(s\,..., sn) e Sn : for some i ^ j, SiSj > 0}.
Proof: The proof of the necessity is identical to that in Theorem 4.6.1. To
establish sufficiency, note that (4.6.5) implies that T-almost every vector s =
(si,..., sn) € 5+ has at most two of its coordinates different from zero. Thus, if
Eij is the subset of Sn, where only Sj and Sj are different from zero, and if Dj is
the subset of S„ where only Si is different from zero, then
n —1 n
(Xu...,Xn) ± ]T Y,(Xl(i,j),...,Xn{itj)) + {Wl,...,Wn), (4.6.6)
i=i j=t+i
where, using the notation introduced in the proof of Theorem 4.6.1,
Xk(i,j)= / skM(ds), k= l,...,n; i = l,...,n- 1; j = i+ l,...,n,
and
Wk= SfcM(ds), fc= l,...,n.
Note that the random vectors appearing in the right-hand side of (4.6.6) are all
independent, so that we will be done once we prove that each one of these random
vectors consists of negatively associated random variables. The latter is trivially
true for the random vector (W\,..., Wn), which consists of independent random
variables. Now, for each fixed (i,j), Xk(i, j) = 0 a.s. if k g {i,j}, which
reduces our task to showing that Xi{i,j) and Xj(i,j) are negatively associated.
But (4.6.5) implies s^Sj < 0 T-a.e. on Eiyi, so that, by Theorem 4.6.1, Xi(i, j)
and-X,(i,j) are associated. Thus (see Exercise 4.27), Xi(i,j) and Xj(i,j) are
negatively associated. The proof of the theorem is now complete. I
4.7 The codifference for stationary SaS processes
Let X = {X(t) = JE f{t, x)M{dx-jf -t~€- K} be a (strictly) stationary SaS
process, i.e., for any d > 1, t\,..., td € R, the finite-dimensional distribution of

4.7 THE CODIFFERENCE FOR STATIONARY SaS PROCESSES 209
the vector (X(t\ + h),.. .,X(td + h)) does not depend on the choice of h € R.
When the process X is Gaussian, the autocovariance function EX(t)X(Q) totally
describes the dependence. When X is SaS with 0 < a < 2, covariances do not
exist. One could use the covariation but this quantity is not always defined for
a < 1. Here, we shall use the codifference defined in Section 2.10. Since X is a
stationary process, the codifference between X(t) and X(0) equals
= 2\\x(o)\rQ-\\x(t)-x(o)ra
= 2||/<0, .)||S-||/(t,-)-7(0,-)HS, (4.7.1)
where \\f(t, 0||£ = JB \f(t,x)\am(dx) and where m denotes the control measure
ofM.
When comparing codifferences (see Property 2.10.6), it is convenient to
standardize the process (i.e., consider X(£)/||X(0)||Q) or, equivalently, to consider
the standardized codifference
r(t) =2 \\X(t)-X(0)\\°Q
\\x(o)\\aa iraiis "
Example 4.7.1 We shall compute the codifference r(i) for the Ornstein-
Uhlenbeck process
X{t) = f e-xit~x)M(dx), -co < t < oo,
J — oo
defined in Example 3.6.3. Recall that A > 0 and M is a SaS random measure
with Lebesgue control measure. We have ||X(0)||£ = J^ooeaXxdx = -^ and,
for t > 0,
||X(t)-X(0)||S = / eaXx\e-xt-l{x<0)\adx= f +f
J — oo J—oo JO
1 '- -Xt\a , * ft „-aXt\
so that
= -^-(1 - e~xt)a + -i-(l - e~aX%
aX aX
- T(t) = -L[i - (1 - e-xt)a + e-aXti t > 0.
aX
The standardized codifference equals
T(t)
l|A-(0)||2
In particular,
_ i_(i_e-*«)0+e-aAt, t>0.
T(t) = \e~xt, if a = 2
A

210
DEPENDENCE STRUCTURES
4.7
(which is the autocovariance function of the Gaussian Ornstein-Uhlenbeck
process),
2
T(t) = -re~xt, ifa= 1
A
(this is the codifference of the Cauchy Omstein-Uhlenbeck process) and, as t —►
CO,
if 1 < a < 2,
r(t)
~ <
i„-At
Ae
-At
ifa= 1,
^ Z\e~aXt ifO<a< 1.
The standardized codifference T(t)/\\X(0)\\% = a\r(t) equals 2e~Xt for a = 2
and a = 1, and as £ —► oo,
r(t)
ae
-At
if 1 < a < 2,
ll*(0)H2
~ <
2e~At ifa=l,
e-«At ifo<Q<i.
As a function of a, the asymptotic expression for the standardized codifference
decreases as a goes from 0 to 1, is discontinuous at a = 1, and then increases as
a goes from 1 to 2. But, for any fixed 0 < a < 2, and for all t > 0,
ta(*)<tv(*) ifA>A'>0,
where ta(£) and tv(4) are the rs that correspond to the standardized SaS
Ornstein-Uhlenbeck processes X\ and X\i with parameters A and A',
respectively. Therefore, by Property 2.10.6, if A > A' > 0,
P{\Xx(t) - Xx(0)\ >c}> P{\Xx.(t) - Xy(0)\ > c} for all c > 0,
i.e., (Xy{t), Xa'(0)) is "more dependent" than (Xx(t), Xx(0)) for all t > 0.
Samorodnitsky and Taqqu (1994) show that the following relation holds
for standardized SaS Ornstein-Uhlenbeck processes {Xx(t), t € K} and
{Xx>(t), teR} with 0 < a < 2: if A> A' > 0, then, for all t,,... ,td eRand
all real numbers c\,..., Cd,
P{Xx(U) > c,.. .,Xx(td) > cd} < P{Xx'(tx) > cu.. .,Xx>(td) > cd}.
(4.7.2)
This last relation has the following interpretation: the probability that the
components of Xx are all large is not greater than the corresponding probability for Xy ■
Hence, the components of Xy are "more dependent" than those of Xx-

4.7 THE CODIFFERENCE FOR STATIONARY SaS PROCESSES 211
The inequality (4.7.2) holds also for a = 2, but in that case, it follows
from the Slepian conditions EX\(U) = EX\,(U) for every i = \,...,d and
EX\(ti)X\(tj) < EX\'(ti)Xy(tj) for every i,j = 1,..., d (see, for example,
Adler (1990) or Ledoux and Talagrand (1991)). The Slepian conditions hold for
X\ and X\i because these processes are standardized and EX\(ti)X\(tj) =
2exp{-A|ii-i/|}.
We shall be mostly interested in the asymptotic behavior of r(t) as the lag t
tends to infinity. We saw in the previous example that the codifference r(t) for
the Ornstein-Uhlenbeck process tends to zero as t —> oo. This, in fact, always
happens if the process is a SaS stationary moving average, i.e., a process of the
form
/oo
f{t-x)M(dx), teR,
-oo
where M is a SaS random measure with Lebesgue control measure. To prove it,
we need the following lemma.
Lemma 4.7.2 Let f,g € La(E,£,m), 0 < a < 2. Then
jj|/(z)r - \9(x)\a\m(dx) < C ^ \f(x) - g(x)\am(dx)j
(4.7.3)
with
1 if 0<a< 1,
21/Qa(||/||r' + Hsllr') if Ka<2.
Proof: Suppose, firstly, 0 < a < 1. Then \x + y\a < \x\a + \y\a, and thus
|a:|« = \x-y + y\a <\x- y\a + \y\a, yielding ||x|a - \y\a \<\x- y\a. Hence,
jE ||/|« _ \g\<*\dm < JE |/ - g\adm, and so (4.7.3) holds with C = 1.
Suppose, now, 1 < a < 2. The mean value theorem applied to |x|a gives
||x|<* _ |y|aj < Qjx _ ylllsl01-1 + |y|a~'|- Hence, using the Holder inequality
(with conjugate exponents a and a/{a — 1)) and (2.7.6), we obtain
[\\f\*-\g\a\dm
JE
<a [ \f-g\(\f\a-l+\g\a-l)dm
JE
<a[J \f - g\adm)l/0(jE(\f\a~l +\9\a-l)^dm)~
<a(/ \f-g\adm)i/a 21^-1)(^|/rdm + j^dm)

212 DEPENDENCE STRUCTURES 4.7
< (J 1/ - 9\adm) Wa2l'°a(\\f\\r' + \\g\\%-1). I
Theorem 4.7.3 For a SaS, 0 < a < 2, stationary moving average process,
lim r(t) = 0.
PROOF: The moving average X is well defined if and only if ||-X"(0)||£ =
JT \f(x)\adx < oo. Suppose, firstly, that / has compact support. Since, for
large enough t, the supports of f(t + •) and /(•) are disjoint,
/oo
\f(t + x)-f(x)\adx
•oo
/oo /»oo
\f(t + x)\adx+ / \f{x)\adx
= 2||X(0)||2
and hence limt-»co T(t) =0.
If / does not have compact support, let e > 0 be arbitrary and choose a
bounded interval Kt such that /^ |/(£)|Ql/src(x)<i£ < e, where l#e is the
indicator function of the complement of Ke. Now choose t large enough so
that the functions /(•)!*«(•) and f(t + -)\kM + ') nave disjoint support. For
such t, r€(t) = 0, where t£ is t with / replaced by }\k,- Moreover, letting
/t(-) = /(« + 0 and (/l^.)t(-) = /(* + -)!«■«(* + ■)• we see bV Le"™3 4-7-2-
that
|r(t)| = |T(t)-r«(t)|
/oo />oo . I
\\f\a-\f\Kt\adx+ / I/* -/r - K/iicj* -/i/c.rkc
-oo' •/— oo1
/CO rOC |tt -)1A(1/Q
1/1^1°^+C/. /t-(/lK.)t-(/-/lif.) dx\
-oo •/—oo
/oo r /-oo /«oo - iA(i
\f\Kc\adx+\2C \(f\Kc)t\adx + 2C \f\KTdx\
-co •■ J—ca J — oo J
<2£+(4Ce)1A(1/Q). I
In the literature one can also find two quantities, parametrized by —oo <
6\, 62 < oo, which are closely related to the codifference r(t). The first is
U{6ue2\t) = EeWXto+WW - Eeie'x^Eei92XW, (4.7.4)
)
/a)

4.7 THE CODIFFERENCE FOR STATIONARY SaS PROCESSES 213
i.e., the joint characteristic function of (X(t),X(0)) minus the product of the
marginal characteristic functions. The second is
I{9u92;t) = -lnSe^W+^W) + lnEeie'x^ + \nEeie^°\ (4.7.5)
which is related to U{9\, 92; t) by
U(eu62;t)-K(eu92)(e-I^W-\),
where K(9U92) = {Eexpi9xX(t))(Eexpi92X(0)) does not depend on t since
X is stationary. If I(9i, 92; t) —> 0 as t —* oo, then
U(9{, 92; t) ~ -K(9l, 92)I(9l ,92;t), (4.7.6)
i.e., U and I are asymptotically proportional.
Both U and I can be used if the process is skewed a-stable. If the process is
SaS, then
T(t) = -J(l,-l;t). (4.7.7)
Remarks
1. Iflimt_oo-f(0i,02;t) = Oforanyfli and 92, then the process {X{t), t e R}
is mixing, that is,
lim
t—*oo
P{A n B) - P{A)P(B)
0
for every A S cr{X(s), s < 0} and B € a{X(s), s>t} (cf. Maruyama
(1970) and Gross (1993)).
2. Gross (1993) also shows that for stationary SaS stable processes satisfying
Condition S, the relation "lim^oo I(9u92;t) = 0 for all 0t and 82" (and
thus that {X(t), t 6 M} is mixing) follows from limt-.^ t(*) = 0 when
0 < a < 1, and from lim^oo r{t) = 0 and limt_QO 1(1,1; t) = 0 when
1 < a < 2.
The advantage of I over r stems from the presence of the parameters 9\ and
92. Consider, for example, two standardized stationary SaS stochastic processes
X = {X(t), t e E} and X' = {X'{t), t e E] and let r and I relate to X and
t' and V relate to X'. One way of showing that X and X' are different processes
is to show that r(t) is not asymptotic to r'(t) as t -> oo. If r(t) is asymptotic
to r'(£), one can then turn to I and /': the processes X and X' are different if
I(9i,02;t) is not asymptotic to r(9i,82;t) as t -> oo for some 0i and 02- We
now illustrate this technique.

214 DEPENDENCE STRUCTURES 4.7
Suppose that {X(t), t £ E} is a stationary sub-Gaussian process, i.e.,
X(t) = Ax'2G(t), -oo<t<oo,
where {G(t), -co < £ < 00} is a stationary Gaussian process with variance
a\ = Var G(0) and autocorrelation function p(t) = EG(t)G(0)/a%. The vector
A is independent of the process G and is distributed as in Relation (2.5.1). (Sub-
Gaussian processes are defined in Section 3.7. See also Section 2.5 where sub-
Gaussian random vectors are introduced.) Since
£ei(«iJf(W(0)) = exp.
(see (2.5.4)), we have
2 Var(0,G(i)+fcG(O))
a/2"
ll<«o)||2 = -
Var^GCi))
. a/2
and
Hence,
||0,*(t) + e2x(o)\\a = (y)Q/V? + 022 + 2e,^P(t))a/2.
>q/2.
,Q/2 r.
i(0i,fc;0 = (y)Q [(02 + 922 + 2e1e2P(t))^2 - |e,|° - |fcp]. (4.7.8)
Setting r(t) = —7(1, — l;t), we obtain
Proposition 4.7.4
Ax'2G{t), t e R}(
For the stationary sub-Gaussian process {X(t) =
T(t)
(£»W2
^ 2 ;
2 - (2 - 2p(t))a/2
, te:
(4.7.9)
//, m addition, limt_oo p(i) = 0, then
T2. a/2
lira r(t) = (Q) (2 - 2Q/2) > 0 /or 0 < a < 2.
Let us now prove that stationary SaS moving averages are different from
stationary SaS sub-Gaussian processes when 0 < a < 2. Let X(-) = All2G(-)
denote a stationary SaS sub-Gaussian process and let Y(-) be a stationary SaS
moving average. Let p(t) be the corelation of G. Since limt_oo ty (£) = 0
(Theorem 4.7.3), it is sufficient to show that Tx(t) (given in (4.7.9)) does not
converge to 0 as t -* oo. Suppose, to the contrary, that limt-,oo tx (t) = 0. Then
limt-oop(t) = Poo where 2-(2-pco)"/2 = O.thatis.poo = l-2<2/a>-'-. Since
Poo < 0, the correlation function of G cannot be non-negative definite (Exercise
4.31), which is a contradiction. We have therefore established

4.8 EXPECTED NUMBER OF LEVEL CROSSINGS 215
Theorem 4.7.5 Stationary SaS moving averages are different from stationary
SaS sub-Gaussian processes for any 0 < a < 2.
We now turn to the SaS sub-stable processes {X(t) = Al/a'Y(t), t £ R},
defined in Section 3.8. Here, Y is a Sa'S stationary process and 0 < a < a' < 2.
Let tx (t) and Ty (t) denote, respectively, the codifference for the processes X and
Y. Since || £j=, bjX^Wl = || £j=, b^t^, (Relation (3.8.2)), we have
rx(t) = 2||y(o)||°, - \\Y(t) - y(o)|fr. But 7y(t) = 2||y(o)||«; - \\Y(t) -
y(0)||°; and, therefore,
Proposition 4.7.6 For the stationary sub-stable process {X (t) = AllaY(t), t €
rx(t) = 2||y(0)||^ - (2||y(o)||s: - Mt))a/a'.
If Ty (t) —+ 0 as t —► co (e.g. if Y is a moving average), then
limrx(i) = ||y(0)||S-(2-2Q/"')>0.
t—*oo
If, moreover, X is standardized (||X(0)||Q = 1), then ||y(0)||Q/ = 1 and
limt^oo tx{t) = 2 - 2a/a which is different for different a'. This shows,
for example, that SaS sub-stable processes with different a' are different (if
rY{t) - 0).
4.8 The expected number of level crossings for
stationary sub-Gaussian processes
Let {X{t), t > 0} be a SaS sub-Gaussian process, i.e., {X{t) = Al/2G{t), t >
0} defined in Section 3.7. We want to evaluate the mean number of crossings
of a given level usEby the process X. We shall assume that {G(t), t > 0}
is a mean zero stationary Gaussian process so that X is stationary as well. The
autocovariance function R(t) = EG(t + s)G(s) of G can be expressed as R{i) =
ST',*, eiXtdS{\) where S is the "spectral distribution function" (see, for example,
Brockwell and Davis (1991)). Observe that the variance iJ(0) = /!^,dS(A)
equals the total spectral mass. IfHisdifferentiableatzero,thenfl'(0) = 0 because
R is even. If R is twice differentiable at zero, then -R" (0) = J"^ X2S(dX) > 0.
The quantity -R"{0) is called the second spectral moment. We will assume in
the sequel that -i?"(0) exists. 3
Let CU{T) denote the number of crossings of the level u by the SaS stationary
sub-Gaussian process {X{t) = Al/2G(t), t > 0} during the time interval [0,T].
3We also assume that {G(t), t > 0} has continuous sample paths. The existence of R'{0) is
sufficient to ensure that such a version exists (Leadbetter, Lindgren & Rootzgn 1983, p. 152).

216
DEPENDENCE STRUCTURES
4.8
(For a formal definition of level crossings see, for example, Cramdr and Lead-
better (1967) or Leadbetter, Lindgren and Rootzdn (1983).) We want to evaluate
ECU(T). (By symmetry, ECu(T)/2 equals the mean number of upcrossings.)
Since X is stationary, it is immediate that ECU(T) = TECU(\), and so from
now on we shall only study Cu := Cu(l).
The mean number of crossings EC^ of the level u by the Gaussian process
G is given by the Rice formula
(Rice (1945)). This formula shows that EC2 depends only on the variance R(Q)
and on the second spectral moment -R" (0) of G. If we denote the distribution
function of A by Fa/2, then it is immediate from the relation X(t) = A,/2G(t)
and Rice's formula that for a stationary sub-Gaussian process,
Clearly, ECU is a monotone decreasing function of |u|. Since, however, Fa/2
is not generally known, (4.8.2) will not usually enable us to obtain an explicit
formula for ECU. In one special case of interest, the Cauchy case a = 1, one can
obtain ECU explicitly because F\/2 has the density
e-l/4o
/l/2(a) = VW
(see (1.1.15)). Substituting this into (4.8.2) and performing the integration leads
to
Proposition 4.8.1 IfX is a stationary SIS sub-Gaussian process, then
ECu= V^W)
V#(0)+2u2'
Proof: Exercise 4.32. I
Observe that ECU is asymptotically proportional to u~l in the Cauchy case
q = 1. We shall now derive an asymptotic formula for ECU for general sub-
Gaussian SaS processes.
Theorem 4.8.2 For a general sub-Gaussian SaS process, we have
hm-U ECu ~ 7rr(l-a/2)(J?(0))(-^) • (4-83)
U—►DO

4.9 EXERCISES 217
Proof: We start with (4.8.2). Since A ~ 5Q/2((cos *f )2/q, 1,0), by Property
1.2.15,
lim aa'2{\ - Fan{a)) = lim aal2P{A > a)
a—>oo a—oo
1 — a/2 7ra
= - rrw
r(2-a/2)cos(7ra/4) 4
= (r(l-a/2))-'.
It therefore follows that
/•OO
Junov0,/2 / exp(-v/a)Fa/2(da)
JO
= lim va/2E(e-v/A)
V—»0O
= lim va/2 / P{e~v'A > x\dx
v^°° Jo
= lim / va/2P{A>v/t}e-tdt
v->ooJQ
/•OO
= / lim va/2P{A > v/t}e_tdt
Jo "-*°°
T(l-a/2)J0
= r(l+a/2)
r(l-a/2)'
where the interchange of limit and integration is justified by dominated
convergence since va/2P{A > v/t} is bounded by ctal2 for some finite c.
Substituting the above into (4.8.2) proves the theorem. I
Whereas ECU decreases exponentially fast in the Gaussian case, it is
asymptotically proportional to u~a in the sub-Gaussian case with index 0 < a < 2.
4.9 Exercises
Exercise 4.1 Let M be a SaS random measure on (E,£), 1 < a < 2, with
control measure m. Show that
E{M{A2)\M[Al)) = m(^f2)M(At) a.s.
m{A\)
Exercise 4.2 Let XuX2be jointly SaS with 1 < a < 2 and let Y\ = aXi + 6X2
and Y2 = cXx + dX2. Find the constant k such that E(Y2\Y\) = fcYj a.s.

218 DEPENDENCE STRUCTURES 4.9
Exercise 4.3 Show that the jointly SaS random variables (A"i, X2, X3, X4)
defined below are such that X\,X2,X^ are linearly independent but the system
(4.1.6) does not have a unique solution.
Here Xi = J0 fi(x)M(dx), i = 1,2,3,4, M has Lebesgue control measure,
and
/,(s)is 1 ifie (0,-), is -1 if ace (-,-), isO if 3; e (-,l);
/2(x)is 1 ifi€ (0,-), is -lifx€ (-,l);
/3(x)is 1 ifxe(o,i), _2'-Qifa;e(i,|), -4'-ttifxe(^,i).
Exercise 4.4 Let X\, X2, ■ ■ ■, Xn be random variables whose first moments exist
and with joint characteristic function <•>. Show that
E(Xn\Xu... ,X„-\) = a]Xi H 1- anXn a.s.
if and only if for all 6y,..., 0n_ 1,
-~z~<j>{0\,...,6n-\,8n)
= ^ — </)(01,...,0n_1,O) + ----(-an_1— ^i,...,fln-i,0).
Exercise 4.5 Use Exercise 4.4 to show that E(Xn \X],..., Xn- 1) is always linear
when (X\,..., Xn) is sub-Gaussian with 1 < a < 2.
Exercise 4.6 Use Exercise 4.4 to show that if X\,..., Xn are jointly SaS, 1 <
a < 2, with spectral measure I\ then
E(Xn\Xu- ■ ■,Xn-i) = oi-^i + • • • + an-iA"„_, a.s.
if and only if for all B\,..., 0n-\,
f (sn - aiSi On_i«„_i)(0iSi + • • • + fln-^n-O^-^nds) = 0.
Jsn
Hint: See Miller (1978).
Exercise 4.7 Suppose that (A*i, X2, X3) are jointly SaS with 1 < a < 2. Use
Exercise 4.6 to show that when E{X3\Xi, X2) - axX\ + a2X2, the coefficients
ax and a2 satisfy the system (4.1.6). Solve that system when A"i and X2 are not
multiples of each other.

4.9
EXERCISES
219
Exercise 4.8 Let c be a real number and X\ and X2 be random variables. Show
that the relation (X\,X2 — cX) — {X\,cX\ — X2) is equivalent to {X\,Xi) =
(Xu2cXi-X2). .
Exercise 4.9 Let (X\, X2) be SaS, 0 < a < 2, with spectral measure I\ Show
that the relation (Xi, X2) = (Xi, 2cX\ - X2) is satisfied if T is given by (4.2.4),
but it is not satisfied if T is given by (4.2.5).
Exercise 4.10 Fill all the details in the proof of Proposition 4.3.2.
Exercise 4.11 Prove that the right-hand side of Relation (4.4.3) coincides with
the right-hand side of Relation (4.4.1).
Hint: Make a change of variables from Sn to E and use (3.2.4).
Exercise 4.12 Prove the second part of Lemma 4.4.2.
Exercise 4.13 Let X be a positive random variable, independent of an exponential
random variable T\ with mean 1. Then
lim A-'P(Xr, < A) = E{X~l).
Exercise 4.14 Let X > 0 and -co < p < 00. Show that EXP < 00 implies
lhm-^ X>P{X > A} = 0.
Exercise 4.15 A gamma random variable TT with parameters r > 0 and 7 > 0
has density function f(x) = -f£pjXr~le~~>x for x > 0 and f(x) — 0 for x < 0.
Show that for any- -00 < p < 00:
(1) £(rY)P = ooifp<-r
(2) Ifp> -r, then
E(Try - 7-p^TTT^ - CrP'
r{r)
where C = C (7, r,p) is a positive constant.
(3) If0<a<2,
E(Ti)-^^ = 00 and i < [-1 and E^r12^ < Ci~2/a fori >
+ 1.
Exercise 4.16 Let Tj be a gamma random variable with mean ETj = j. Show
that
nm^ooSexpO-'7"^ - j)} < 2ex'2.

220
DEPENDENCE STRUCTURES
4.9
Exercise 4.17 Prove that for A > 0 and 0 < e < 1,
P( min (Xk + Yk)>\)
< p({ min Xi>X(l-e)}u{ max Yk > A(l - e)}
\ fe=l,...,n fc=l,...,n
U { max Xk > Ae} U { max yfc > Ae}).
k—l,...,n fc=l,...,n /
Exercise 4.18 Prove Lemma 4.4.4 by induction.
Exercise 4.19 Prove Corollary 4.4.6.
Exercise 4.20 Prove Relation (4.4.15).
Exercise 4.21 Let X\,..., Xn be non-negative random variables. Prove by
induction that
roo too
E(Xi---Xn)= ... P(Xl>tl,...,Xn>tn)dtl...dtn.
Jo Jo
Exercise 4.22 Let 0 < a < 2. The spectral measure can be used to express the
dependence structure of a-stable random variables X\,Xi,..., Xn.
(1) Show that the XjS are independent if and only if T is concentrated on the
2n points (0,..., 0, ±1,0,..., 0).
(2) Show that the XjS are n-fold dependent if and only if
T{se Sn : sis2...sn^0} >0.
(3) Find a necessary and sufficient condition in terms of T for
£|X,p»...LYn|p" <oo.
Exercise 4.23 Show that if U\ and C/2 are associated random variables, then
P(Ui >Xi,U2> A2) > P(*7, > Xi)P(U2 > A2).
Exercise 4.24 If two random variables are associated and their covariance is
zero, then they are independent.
Hint: Prove it first for positive random variables.
Exercise 4.25 If T\,T2,..., Tk are associated random variables and if Gi =
gi(T\,T2,...,Tk), i = l,...,n, where for each i, the function gi is non-
decreasing in each of its arguments, then the random variables G\,G2, ...,Gn
are associated.

4.9
EXERCISES
221
Exercise 4.26 Show that neither F-positive orthant dependence nor F-positive
orthant dependence imply association.
Exercise 4.27 Show that if X and —Y are associated, then X and Y are negatively
associated.
Exercise 4.28 Show that for all 0 < a < 2,
J(l,-l;i) + I(l,l;i)<0.
Hint: Proof of Lemma 2.7.14.
Exercise 4.29 Compute t(£) for the reverse Ornstein-Uhlenbeck process (defined
in Example 3.6.4).
Exercise 4.30 Show that I{0U92; t) (defined in Relation (4.7.5)) tends to 0 as t -*
oo for all #i and 92 if the process is a stationary SaS, 0 < a < 2, moving average.
Show that this is also the case if the measure M in X(t) = J^ f{t — x)M(dx)
is an a-stable (not necessarily symmetric) random measure.
Exercise 4.31 Let {G(t), t £ E} be a stationary Gaussian process with
autocorrelation function p(t) converging to p^ as t —» oo. Prove that p^ cannot be
negative4.
Hint: Suppose p,^ < 0 and choose time points t\,...,tn separated by long
enough intervals so that the correlation matrix of the vector (G(ti),..., G{tn))
consists of 1 on the diagonal and all off-diagonal entries are less than p^ + c < 0.
Choose n sufficiently large to make all the entries of the correlation matrix add up
to a negative number.
Exercise 4.32 Prove Proposition 4.8.1.
4Suggested by Aaron Gross.

Chapter 5
Non-linear regression
We proved in Chapter 4 that the bivariate regression is linear when the random
variables are jointly SaS and 1 < a < 2. In this chapter, we give a complete
picture of bivariate regression in the general, possibly asymmetric, case. We show
that when skewness is present, regression can be either linear or nonlinear. We
also show that the regression can be denned even when 0 < a < 1. Section 5.5
summarizes the main results of this chapter.
In Section 5.1 we give sufficient conditions for the finiteness of the conditional
moment £(|X2|p|Xi = x) when a < p < 2a + 1. When p < a, the conditional
moment always exists because E\X2\P < oo. However, the conditional moment
can also exist when p > a. This depends on the spectral measure T of the a-stable
random vector (X\, Xz). T is a measure on the unit circle. The smaller the mass
concentrated around the north and south poles of the unit circle, the higher the
value of p. These results are relevant to the regression problem. Setting p = 1,
we obtain a sufficient condition for the existence of the regression in the case
0 < a < 1. This condition will become our standard assumption.
In Section 5.2 we derive the form of the bivariate regression. The formulas
are valid for 1 < a < 2, and also, when the standard assumption holds, for 0 <
a < 1. We give necessary and sufficient conditions for linearity. Interestingly,
the regressions are typically asymptotically linear. In Section 5.3 we apply the
bivariate regression results to important classes of a-stable laws, namely those of
moving averages, sub-Gaussian and harmonizable vectors. The sometimes exotic
behavior of the regression functions in the non-linear case is illustrated graphically
in Section 5.4.
The regression in the non-linear case involves two integrals which cannot
be computed analytically. To make matters worse, these integrals have features
which render them intractable by standard numerical integration procedures. In

224 NON-LINEAR REGRESSION 5.1
Section 5.5, we give algorithms for computing the regression formulas. Since a
straightforward application of the midpoint trapezoidal or Simpson's rule gives
a very poor approximation, we developed variants with variable step size and
cutoffs. Although we make no claim that the procedures described here represent
the best possible way to compute the regressions, they are the most appropriate
among those examined. To facilitate applications of the regression formulas we
rewrite them in Section 5.5 using the integral representation. We also refer to
a software package for computing bivariate regressions and probability density
function of a-stable random variables. A description of the package and the
listing of the source code, written in the C language, can be found in Hardin,
Samorodnitsky and Taqqu (19916).
5.1 Conditional moments of order greater than or
equal to a
Let (X\,Xi) be an a-stable random vector. Clearly, the conditional moment
E(|Z2|p|Xj = x) is finite for almost every x if p < a because £|X|P < oo.
What about p > a? In this section some partial answers are provided.
It is well known that the existence of moments is related to the behavior of the
characteristic function around the origin. We shall therefore consider the behavior
of the conditional characteristic function (t>x2\x(r) — E[e%rX:! \Xi = x] as r —» 0.
We consider, firstly, a d-dimensional a-stable random vector X =
{XltX2,...,Xd) and study
4>xd\x x„_,(r) = £[eir**|X, = xu .. .,Xd_{ = xd-X).
Let#x(£i, • • • ,td-\,r) = £exp{i(£ ~j tjXj+rXd)} denote the characteristic
function of the vector X and let T be its spectral measure. Recall that T is a measure
on the unit sphere Sd of Rd. We start with the following simple result.
Lemma 5.1.1 Ifd>2 and for each (f i,..., id-i) € Sd-\,
r{s€Sd:Us1 + --- + td-lsd_l ^0}>0, (5.1.1)
then for each real r,
I |0x(ti,...,*«f-i,r)|dt, ...dfc-! <co. (5.1.2)
JUd-l
Remark. Note that S\ consists of the two point set {—1,1}. Hence Relation
(5.1.1), in the case d = 2, is equivalent to T{(s\, S2) S S2 : S\ ^ 0} > 0, i.e., to
r{s2\{(o,i),(o,-i)}}>o,
(5.1.3)

5.1
CONDITIONAL MOMENTS
225
that is, to X\ ^ 0 a.s.
Proof of Lemma 5.1.1: Define M : Rd~l -> K+ by
M(t,J...,td_1)= / |tia, + -• • + td_isd-irr(«fa).
Jsd
Then
M0 = inf M{tu...,td-i)>0.
(ti,...,td-t)esd-i
IfO<a< 1, then, by (2.3.1),
\4>x.(U,---,td-i,r)\
= expj- / \tysi + htd-iSd-i +rsd|Qr(<is)|
< exp{- J (|t,s, + • • • + t«£_,ad_,r - |rS(i|Q)r(ds)}
< ecMaexp{-M0(t? + --. + <5_1)Q/2},
with C > 0, proving (5.1.2). The case 1 < a < 2 is similar. I
Fix d > 2 and let Fx xd_,,xd, -Fx,,...,xd_, and
Fxd\xi,...,xd-\ denote,
respectively, the distribution function of (Xi,..., Xd- i, Xd), (X[,..., Xd_ i)
and the conditional distribution function of Xd given X\ = xl:... ,Xd-\ = xd-\-
Further, let
/oo
eirXddFxd|x,,...,Id_,(xd)
-oo
denote the corresponding characteristic function. We have by Fubini's theorem
<f>x{t\,..-,td-\,r)
= f eiit>x>+-+t*-<x*->+rx-')dFXu...tXd_l,Xd{xl,...,xd-l,xd)
= / ei(t'x'+-+td-'x"-'^xd|x, *<_,(r)dl;xI,...,x<1_1(x,,...1id-i).
Therefore, for fixed r, we may regard 4>x (* l > • • •. U-1. r) as the Fourier transform
of a complex-valued finite measure Gr on Md_1 given by
(A) = J <pXdlxi xd_,MdFX|,...1x(l_l(si>...,a:d-0 t5-1-4)
/A
for any Borel set /I in Rd_1.
Suppose, now, that (5.1.1) holds. Then both Gr and Fx, xd_, have bounded
continuous densities. Indeed, by Lemma 5.1.1, 0x(ii, • • • >*d-i>r) is absolutely

226
NON-LINEAR REGRESSION
5.1
integrable in t\,... ,i<f_i, so that, by Fourier inversion theorem in Md_1, Gr is
absolutely continuous with respect to Lebesgue measure in Ed-1, with a bounded
continuous density gr{x\,..., Xd-\) equal to
_L_ f e^*>+-"+*'->*'->)fa(tl,...,U-Ur)dtl...dtd-l. (5.1.5)
(27r)a yRd-i
Setting r = 0 in (5.1.5) shows that Fxu...,xd-, also has a bounded continuous
density fx,,...,xd-, with respect to Lebesgue measure on Md_1 equal to
—L^[ e-i(t'a!'+-+f--'as--|)^x(tI,...,i(,_,,0)dt1...did_I. (5.1.6)
(27T)a ' JUd-,
We conclude from (5.1.4) that
gr(xu...,xd-i) = fx ,xd_,(xi,...,x(f_i)0xd|xl,...,id_,(f).
Hence, for almost all realizations X\ = x\,.. .,Xd-] = Xd-\ corresponding to
/x,,...,xd_,(a;i,---,a:d-i) > 0,
, , >. 9r(x\,-..,xd-i)
<PXd{xu-,Xd-,{r) = """"
(5.1.7)
fx,,..
_ f*-> e-^'"+-+t-.-'--)^x(f i,..., td-u r)dtx ... dtd_,
Jr.-, e-*(*>*>+-+t*-«iB--')^x(t,,...,td-uO)dU ... dtd-i
Now that we have obtained an expression for the conditional characteristic
function (j>xd\x,,...,x<i-,< we can exploit the well-known fact that nniteness of
moments of a probability law is directly related to the behavior of its characteristic
function at the origin.
Theorem 5.1.2 Let F be a distribution function and 4> the corresponding
characteristic function.
(i) A necessary and sufficient condition for F to have the moment of order p,
where 0 < p < 2, is that, for some c > 0
1 - Re <j>(r)
f
Jo
o r'+P
-dr < oo.
Moreover, F has a second moment if and only ifr 2 (1 — Re <j)(r)) /' s bounded
for 0 < r < c.
(ii) A necessary and sufficient condition for F, having the moment of order 2n,
to have the moment of order 2n + p, where 0 < p < 2, is that, for some
c>0,
Re0<2n>(O)-Re0(2n>(r)
/
Jo
-dr < co.
»i+p

5.1
CONDITIONAL MOMENTS
227
Moreover, F has the moment of order 2n+2 if and only ifr 2 (Re 0(2n) (0) —
Re ^2n) {r)) is bounded for 0 < r < c.
PROOF: It is enough to prove part (i). Part (ii) follows from part (i) since
0(2") (r) J'(j)(.2n) (o) is a characteristic function corresponding to the distribution G
given by G{dy) = y2nF(dy)/EY2n.
Suppose that /^ \x\pF(dx) < oo for 0 < p < 2 and let c> 0. Now
f l~*Z*T)*r - f ± ^ (1 - cosrx)F(dx)dr
JQ ' JO ' J—oo
because /0°°(1 — cosi)/i1+pdt < oo for 0 < p < 2.1 Conversely, suppose that
J0C(1 — Re <j>(r))/ri+pdr < oo for some c> 0. Then, as above,
oo > r |x|" f * ^-^di F(dx) > [ \x\? f 1—^-dt F(dx)
J-oo JQ l y J\x\>\ JO t P
= const. / \x\pF(dx),
which implies that F has a p!h moment.
Now suppose that F has a second moment. Then cj> is twice differentiable and
,» = lim ^)+0(-r)-2»(O) = _2 Um 1-Re^)
r—»0 T r—*0
Hence, r~2(l - Re <j>(r)) is bounded in the neighborhood of zero. Conversely, if
this quantity is bounded for 0 < r < c,2 then, by Fatou's lemma,
\-Rsd>(r) ,. . r r00 1 -cosra:-, , .
oo > lim inf —-1 = hm inf / ~ F(dx)
r—0 r2 r->o J_00 rz
>/ lim inf = F(dx) = - x'F{dx),
J-oo r^° r 2 7_oo
and the theorem is proved. I
'In fact, one can see that J™ |i|pF(di) = const. Jo°°(l-Re</)(r))/r1+Pdr(cf. also Theorem
11.4.3 in Kawata (1972)).
2The proof implies that it is enough to assume that r ~2 (1 - Re 4>{r)) is bounded at some sequence
of points tending to zero (cf. also Ramachandran and Rao (1968)).

228
NON-LINEAR REGRESSION
5.1
Since we are mainly concerned with regression, i.e., the case p = 1 < 2 (n =
0), we want to study how fast 4>Xd\x .x^O") approaches 1 as r -* 0. When
d = 2, (5.1.7) reduces to
1 f00
fo,|»(r) = 1 + 2nfx (x) J_ e->tx(ct>x(t,r)-4>x(t,0))dt, (5.1.8)
where we set x\ = x. We shall now consider only the case d = 2 and study how
fast the numerator in the right-hand side of (5.1.8) tends to 0 as r —* 0.
Let X = (X\, X2) be a SaS random vector in K2 with a spectral measure T
satisfying (5.1.3). Substituting d = 2 in (2.4.2) we obtain
4>x(t,0) = exp{- j \tst\aT(ds)} = e-l'l"'",
where, by (2.3.3), cti = (/g |si|eT(ds))1/Qf is the scale parameter of the
distribution of X\. We therefore obtain the following expression for the conditional
characteristic function of X2 given X\ — x:
**-w-,+db>£'"~'~,'r'r
x exp{- j (|tei +rs2\a - |ts,|a)r(ds)} - l\dt. (5.1.9)
The conditional moment £(|X2|p|Xi = x) is finite for almost every x if
p < a, but it can be infinite when p > a. For example, let (X\, X2) = (X\, Y +
Z), where (X\,Y) is a SaS random vector, and Z: is a SaS random variable
independentof(X\,Y). (Theautoregressivemodeloforderl,i.e.,X2 = aXy+Z,
is a special case of this.) The presence of the "independent increment" Z ensures,
of course, that E(\X2\?\Xi) = E(E(\Y + Z\"\XuY)\Xi) = oo a.s. if p > a,
because for such p, E(\Y + Z\p\X\ = x, Y = y) = E\y + Z\p = oo. In terms of
the spectral measure r of the vector (X\, X2), this "independent increment" Z is
equivalent to the presence of atoms at the points (0,1) and (0, -1) on the unit circle
S2. Therefore, heuristically, we would expect that the less the spectral measure
T is concentrated around the points (0,1) and (0,-1), the higher the conditional
moments of X2. The following theorem, which is the main result of this section,
shows that these heuristics are essentially correct.
Theorem 5.1.3 Let X = (X\,X2) be an a-stable random vector (not necessarily
symmetric), with spectral measure T, satisfying
I.
sI|-'/r(ds)<oo (5.1.10)
Si

5.1
CONDITIONAL MOMENTS
229
(2,5)
(2,4)
min(a + v,2a+ 1)
' a
1 2
Figure 5.1: The conditional moment of order p is finite in the
shaded region. The picture is drawn with v = 2.
for some v > 0. 77zen £?(|X2|p|Xi = x) < oo for almost every x if
p< min{a + u, 2a + 1}. (5.1.11)
Moreover, if a > 1/2 and v > 2 — a, then £(X||Xi = x) < ex for almost every
x, and if a > 3/2 and u > 4 - a, then E{X\\X\ = x) < oo/or a/masr every x.
The region where (5.1.11) holds is displayed in Figures 5.1 and 5.2.
Remarks
1. "For almost every x" means for all x in the support of the probability density
function of X\. ~
2. Since Js T(ds) < oo, the case v = 0 reduces to the trivial result
E{\X2\v\X\ = x) < oo for any p < a.
3. Applying the theorem with a < 1, we obtain E(|X2||Xi = x) < oo a.e. if
v > 1 - q. l ''■

230
NON-LINEAR REGRESSION
5.1
4 -
(2,5)
(2,2)
0
1 2
Figure 5.2: The shaded regions indicate where
the conditional moment can be finite. Region A is "'•
for v = 0 (no improvement over the unconditional
moment) and region A U B is for v > a + 1.
4. Cioczek-Georges and Taqqu (1994a, 1994c) prove that the condition v >
p — a in Theorem 5.1.3 can be weakened to u > p — a not only for p = 2
or 4 but for all a < p < 2a + 1.
5. Cioczek-Georges and Taqqu (1994b) show that if - /s In |si |r(ds) < oo,
then E[|X2|Q|Xi = x] < oo for almost every x.
The proof of Theorem 5.1.3 is quite technical. To obtain the existence of higher
moments one may have to deal with the second and even the fourth derivative
of <px2\x(r)- Formal computation of the derivatives can yield divergent terms,
and hence special manipulations are needed. Moreover, one needs to bound the
difference of derivatives (or the characteristic function itself) between points 0
and r > 0 by an adequate power of r.
To give a flavor of the proof and to demonstrate some of the basic techniques,
we prove the theorem only in the two following special cases:
0 < a < 1 and \—a<v< 1,
1<q<2 and 0 <u < 2 - a.

5.1
CONDITIONAL MOMENTS
231
The first case ensures the existence of E(\X2 \ \X\ = x) for almost every x and
the second one (when v = 2 - a) covers also the existence of E[X2\X\ — x] for
almost every x when 1 < a < 2. Other cases are considered in Samorodnitsky
and Taqqu (1991a) (0 < v < 1 - a for 0 < a < 1) and Cioczek-Georges and
Taqqu (1994a, 1994c, 1994ft).
Before proving Theorem 5.1.3 in the special cases indicated above, we need
to collect certain technical results. Suppose that (5.1.10) holds for some u > 0,
and let Cu < oo denote the corresponding value of the integral in (5.1.10).
Lemma 5.1.4 (a) Let 0 < a < 1. For any real, non-zero r and t,
JSi
(\tSl+TS2\a - |ts,|a)r(ds) <r(&)|r|a.
(5.1.12)
(ii)
Us2
(\tsl+rs2\a-\ts1\a)r(ds)\
C„2U{1 + a)\r\a+v\t\-', if 0<u<l-a,
< < (5.1.13)
C1_a2,-°(l+a)|r||i|-<,-Q\ if v>l-a.
(Hi)
Js2
\tSl + rs2\a + |ts, - rs2\a - 2|tSl|Q T(ds)
f C„2,+"(a(l - a) + l)Ma+"|ir'/, if 0<v<2-a,
< { (5.1.14)
\ C2-a2>-a{a(l - a) + \)r2\t\-(2-a\ if v>2-ol.
(b) Let 1 < a < 2. For any real, non-zero r and t,
(0
M
Js2
\[ {\t3l+rs2\a-\t3X\a)nda)\<anS2)(\r\a + \r\\t\a-1). (5.1.15)
'J s2 '
\tsl+rs2\a + \tsl-rs2\a-2\ts{\a\nds)
( C„21+^((a - 1) + 3a-')\r\a+v\t\-", if 0 < u < 2 - a,
< { (5.1.16)
1 C2_Q23"aa((Q - 1) + 3a" VM-(2-a), if v>2-a.

232 NON-LINEAR REGRESSION 5.1
Proof: (a) Part (i) does not involve v\ it follows from \\ts\ + rs2\a - \tsi \a\ <
\rs2\a < \r\a- ((5.1.10) is not being used). For parts (ii) and (in), decompose the
whole integral into a sum of integrals J\ and J2 over {{s\, s2) G S2 : l*Si | > 2|r|}
and {(si,s2) G S2 : \ts\\ < 2|r|}, respectively.
To obtain a bound for Jx in part (ii), note that by the mean value theorem,
there is a 0 < 6 < 1
\\tsi + rs2\a - \tSi|a| = \rs2\a |ts, + 9rs2\a~l < |r|a |tsi/2|a-',
since \tsi\ > 2|r| and \8rs2\ < \r\ imply \ts\ + 9rs2\ > |ts,| - \6rs2\ >
|fSl|/2. Hence J, < a21-Q|r||t|-('-Q) /|/ai|>2r |s,|-('-«)r(ds) is bounded by
Cua2v\r\v+a\t\-v if 0 < v < 1 - a and Ci_ctQ21-a|r||t|-(1-«) if 1/ > 1 - a.
Moreover, J2 is bounded above by
/ |rs2|°T(ds) < \r\a [ Istf^-Tids) < Cu2"\r\a+v\t\-»,
J|t3i|<2|r| J\tSi\<2\r\
yielding the first inequality in (5.1.13). For the second inequality, take v = 1 - a.
For part (iii), in the estimate of J\, use the fact that there exist constants
01,02,7^ (0,1) such that
\\tSi + rs2\a + |ts, - rs2|Q - 2|tSl|Q|
= a|ra2||(a - 1)(0. + e2)rs2\tSl + (70, - (1 - 7)02)rs2|a-2|
< 2a(l-a)r2|^r2,
and also use J2 < J|ts,|<2|r| 2|rs2|Qr(ds).
(b) Part (i) follows from the mean value theorem and the proof of part (ii) is
similar to the case 0 < a < 1. 1
Proof of Theorem 5.1.3 (for 0 < a < 1, \ - a < v < 1 and for
1 < a < 2, 0 < v < 2 — a). We first show that if the theorem holds
for SaS random vectors, then it holds for a-stable vectors. Let (X\,X2)
be a-stable, let v and p be as in the theorem and let (Y\,Y2) be an
independent copy of {XUX2). Then (Z,,Z2) J= {XUX2) - {YX,Y2) is a SaS
vector with spectral measure T, defined by T(A) = T(A) + T(—A) for every
Borel set A of S2. Since J^ \si\-"f(ds) = 2jS2 \si\-T(ds) < co, we have
E{E{\Z2\p\XuYuY2)\Zi) = E(|Z2|p|Zi) < oo a.s., since we are assuming the
theorem holds for SaS random vectors. Consequently, E(\Z2\P\X\,Y\,Y2) < oo
a.s. by Fubini's theorem. Since (Y\, Y2) is independent of (X\ ,X2),
E{\X2\v\Xi) = E(\X2-Y2 + Y2\p\XuYuY2)
< 2?(E(\X2 - y2|p| XUYUY2) + E(\Y2\v\ XUYUY2))
= 2"(E(\Z2r\XuYuY2) + \Y2\n

5.1
CONDITIONAL MOMENTS
233
is a.s. finite, establishing the claim.
We suppose from now on that (X\, X2) is a SaS random vector and assume
either 0 < a < 1 with l-a<i/<lorl<a<2 with 0<v<2-ain
the sequel. Observe firstly that (5.1.10) implies (5.1.3), so (5.1.9) holds. Then
decompose the integral in (5.1.9) as follows:
y°° e-it'e-W""? Lp{- J (|iSl + rs2\a - |ts,|a)r(ds)} - 1
= - f° e-^e-l'l'"" [ f (|ts, +rs2\a - |ts,|Q)r(ds)]di
+ f e-it*e~\t\a<>? [_1 + f (|te, + rS2|a _ |iSl|«)r(ds)
J-OO l JS!
+ exp{- f (|ts, +rs2\a - |tsi|a)r(ds)}l<ft
=: -J1+/2.
We are now going to estimate 7,. We have
/oo
costee_|t|Q<Tf
-OO
(5.1.17)
/oo
sin tx e~
-00
=: 7,i-i7,2.
[ (|ts,+rs2|Q-|ts,|Q)r(ds)
[ (Its, +rS2|Q-|ts,|Q)r(ds)
dt
(5.1.18)
Lemma 5.1.4 (Relations (5.1.13) and (5.1.15)) implies that for all 0 < |r| < 1
(say), |7,2| < C\r\. Here and in the sequel, C denotes a positive constant, which
may change from line to line. Moreover, |7,,| equals
Jo
cos tx e
Jo
f (|ts, + rs2|Q + |ts, - rs2|a - 2|ts,|Q)r(ds)
Js2
f ||ts, + rs2|Q + |ts, - rs2|Q - 2|ts,|a|r(ds)
Js2
dt
(5.1.19)
Now use Lemma 5.1.4 to bound the inner integral in the right-hand side of
(5.1.19). Specifically, use the first inequality of (5.1.14) for 0 < a < 1 (note:
v < 1 < 2 - a) and (5.1.16) for 1 < a < 2. We obtain for any 0 < |r| < 1,
|7„|<C|r
\cn-\-u
(5.1.20)
We now turn to the term 72 in (5.1.17). Using the elementary inequalities,
valid for any 0 < y < T,
y-j<l-e-y<y,

234 NON-LINEAR REGRESSION 5.1
V < ey - 1 < y + |"eT,
we conclude, using Relations (5.1.12) and (5.1.15) of Lemma 5.1.4, that |J2| is
bounded above by
C
f
J — c
e-|t|Q<er(s2)|r|-
/ (|fSl+rs2r-|tSlnr(<fc)
Js2
dt
forO < a < 1, and by
c r e-i'r-re^Hirr+Mitr-') \ f (\tSl + rs2\a - |t*,r)r(dB)
j—00 iJ s?
dt
for 1 < a < 2.
In the case 1 < a < 2, we may show that for all 0 < |r| < 1
I/2I < Cr2
(5.1.21)
by using (5.1.15). Consider, now, the case 0 < ex. ^ 1. If 1 /2 <C ex. ^ 1, then
(5.1.13) implies that for all 0 < \r\ < 1,
\h\ <Cr2. (5.1.22)
If a < 1/2, then using both (5.1.12) and (5.1.13), we obtain, for 0 < \r\ < 1,
\h\ < C([ +/ )f/"(|te,+rS2r-|te1|a)r(ds)
l\t\<\r\ J\t\>\r\
< C f \r\2adt + C f
J\t\<\r\ J\t\>r
< C\r\2a+l,
r2\t\2a-2dt
dt
(5.1.23)
If a = 1/2, then consider u1 = (1 - e)/2 < 1 - a with 0 < e < 1 and use the
first inequality of (5.1.13) to obtain, for 0 < \r\ < 1,
|/2| < C\r\2~e. (5.1.24)
We now gather the intermediate results. By (5.1.9), (5.1.17) and (5.1.18),
1 - Re <j>x2\x(r)
= -(27r/x,(a;))-1Re(-J1+72)
= (27r/x,(x))-1(/„ -Re/2) < (2nfXl(x))-l(\Iu\ + \h\).

5.1
CONDITIONAL MOMENTS
235
Hence, by (5.1.21), (5.1.22), (5.1.23), (5.1.24) and (5.1.20),
1-Refo,|x(r)< <
' C(/x,(z))~1|r-|Q+", ifO<a<l,l-a<i/<l
or if 1 < a < 2,0 < u < 2 - a,
. C(/x,(:r))_1|r|2, if 1< a < 2 andi/ = 2 - a,
(5.1.25)
for all 0< jr-j < 1.
Part (i) of Theorem 5.1.2 implies E{\X2\P\X\ = x) < oo a.e. for any
p < a + v in the cases listed in the first inequality of (5.1.25). Moreover, for
1 < q < 2, it implies E(\X2\2\X\ = x) < oo a.e. if v = 2 - a. This completes
the proof. I
Remark. Condition (5.1.10) involves the spectral measure T of the vector
(Xi,X2). Often, however, (XUX2) is given by its integral representation
(Xi,X2) = (/ fi{x)M(dx), J f2{x)M{dx)), (5.1.26)
where M is an a-stable random measure on the measure space {E, £) with control
measure m and skewness intensity /?(•): E —* [—1,1]. It is easy to see by a change
of variables that
where E+ = {x e E : /2(x) + /2(x) ^ 0}. (For a formal proof see Proposition
5.2.10 below.) Hence, Condition (5.1.10) is equivalent to the finiteness of the
right-hand side of (5.1.27).
The following example shows that without additional assumptions (namely,
X2 = aX\), it is not possible to obtain conditional moments of order p > 2a + 1
by merely supposing v > a + 1.
Example 5.1.5 Let Yj and Yi be i.i.d. SaS random variables and set X\ = Y\ + Y2
and X2 = YX-Y2. The spectral measure T of {Xx, X2) is concentrated on the four
points (2'/2,21/2), (-2'/2,-21/2), (21/2,-21/2), (-21/2,21/2) of the unit circle
S2 and hence, in this case, u* = oo, where v* = sup{^ > 0 : (5.1.10) holds}.
On the other hand, let g(x) be the density of Y\ or Y2 and let fXl (x) be the
density of Xx. Since g(x) ~ C|x|-1-Ql as |i| —> oo,
EUXinx, „,,. _i_£,X2l4s(-±£i)s(^)<(lJ
is infinite for p = 1 + 2q but is finite for all p < 1 + 2a.

236
NON-LINEAR REGRESSION
5.2
The following proposition generalizes Example 5.1.5. It shows that one always
has £[|X2|2q+1|^i] = °° f°r linearly independent SaS XUX2, with a discrete
spectral measure T.
Proposition 5.1.6 Let (Xi, X2) be an SaS random vector, 0 < a < 2. Suppose
that there are points s^ ^ ±s^ such that the spectral measure assigns a positive
mass to both. Then
£(|*2|p|A',)=oo
for allp> 1 + 2a .
PROOF: Assign the mass at the above two points with plus and minus signs to
(Yl,r2), all the rest to (ZUZ2). Then {YUY2) and {ZUZ2) are independent
(Proposition 2.3.7), and they sum to (X\,X2). By the argument of Example
5.1.5, £(|y2|p|Yi) = oo a.s. for p > 1 + 2a and, by independence, E(\Y2 +
Z2\P\YUZUZ2) = oo a.s., and so
£(|X2|p|*i) = E{E(\Y2 + Z2\p\Yx,ZuZ2)\Xi) = oo a.s. I
Remark. It seems reasonable to conjecture that, even when T is not concentrated
on a finite number of points, £[|X2|2a+1 |.Yi] = oo unless X\ and X2 are linearly
dependent. Exercise 5.2 illustrates what can happen in the skewed case.
5.2 Analytic representations of the non-linear
regression functions
Let (X\,X2) be a-stable, 0 < a < 2, with spectral representation (T, /x°), i.e.,
characteristic function as in Theorem 2.3.1. ThenX] ~ Sa(cri,pi,p.i), where
a, = (/ |SljQr(ds))I/Q (5.2.1)
is the scale parameter of X\,
0» = -4 / s<Q>r(ds)e[-i,i] (5.2.2)
is the skewness parameter of X\, and
f M? ifa^l,
M. = I (5-2.3)
I ^-|4s.ln|Sl|r(ds) ifa=l,

5.2 ANALYTIC REPRESENTATIONS 237
J&
is the shift parameter of X\ (see Example 2.3.4). In order to study the regression
E{X2\X\ = x), we must ensure that it is well defined. Clearly,
a > 1 => E\X2\ < oo =4> S(|X2||Xi = x) < oo fora.e. Z.
When q < 1, we assume the following:
Condition 5.2.1 lf{X\, X2) is a-stable with spectral measure T and a < 1, then
there is a number v > 1 — a such that
, ?^<oo. (5.2.4)
Ith N"
Theorem 5.1.3 ensures then £(|A'2||Xi = x) < 00 for a.e. x. Observe that the
choice v > 0 is adequate when a = 1 and the choice u = 1 is adequate for all
a < 1.
Since we are interested in the regression E(X2\X[ = x) as a function of x,
we assume a\ > 0, because <7i = 0 implies that X\ is degenerate and hence is
E(X2\XX) - EX2. By virtue of (5.2.1), ex > 0 is equivalent to
r(S2\{(0,l)U(0,-l)})>0. (5.2.5)
We also assume, without loss of generality, p° = (/j,®, $) = 0, i.e., that
(X\,X2) has representation (r, 0), because if fj,° ^ 0, then setting X\ = X\ +$
and X2 = X2 + /x° yields
E(X2\Xl = x) = E(X2\Xt =x-im1) + £,
with (X\, X2) having representation (r, 0).
When n° = 0. *e density function of Xx, fX[ (?) = ^ JZo e~ita:<Ax, (*)rf*.
equals
e^'0 cos(ta-a/?,cr? ta)dt (5.2.6)
if a ^ 1 (where a = tan(a7r/2)), and
fx(x) = ^- [°° e-itxeKp{-ai\t\-0iai-t\a\t\ + iiMXt}dt
27r J-00 ( n
= - /"°° e-ff'' cos (t(x -mi) + -j8iffi* In t)dt (5.2.7)
if q = 1, where /xt = -| /s st ln|s^r(ds).
The following theorem provides an explicit formula for the regression in the
case q^I.
A f°°
f Jo

238
NON-LINEAR REGRESSION
5.2
Theorem 5.2.2 (case a 7^ 1). Let {X\,X2) be a-stable, a ^ 1, with spectral
representation (T, 0). // 0 < a < 1, let (X\, X2) satisfy Condition 5.2.1.
Then, for almost every x.
E(X2\XX = x) = kx +
a(X - 0\k)
l+a2/3?
a(3\x +
\-xH(x)
""/x, (x) J '
where
H(x) = /
Jo
sin (tx - aPicrft0") dt,
k —
_ [X2,Xx}a _Js2s2s<l
<a-l>
r(ds)
JSlg2hla-T(ds)
(5.2.8)
(5.2.9)
(5.2.10)
(5.2.11)
anrf w/iere a = tan ^p, an J o\, (3\ and fxx are, respectively, the scale parameter
(5.2.1), the skewness parameter (5.2.2) and the probability density function (5.2.6)
of the random variable X\.
If a < 1 and fix = 1, Relation (5.2.8) is well defined only for x > 0, and if
a < 1 and (3\ = — 1, /r /$ w// defined only for x < 0.
PROOF: Since (5.2.5) holds, the characteristic function <px(t,r) of X =
(X\,X2) is absolutely integrable with respect to t and therefore the conditional
characteristic function 4>x1\x(t) of X2 given X\ = x equals
**|x(r) = 1 + - / M r e-itx(4>x{t,r) - <^,(t))di.
27T/x, (X) 7.^
(See Section 5.1 and Relation (5.1.8).) Hence, for almost any x,
E(X2\Xl=x) = -»^|x(0)
2^/x, (1) dr
f
J —(
e"ieiG>x(t:r)dt
.(5.2.12)
r=0
We start with the evaluation of g7<£x(i,r)|r=o- For any f ^ 0,
9r
tfx(*,r)
*x(«,0)IimiC$^4-l
r=0
(5.2.13)
r—0 r \0X(t:O)
where <t>x.(t,r)/4>x(t,0) = e-u<t,r> with
"(*.i") = / [|^i+rs2|Q(l-iasign(fsi+rs2))-Sisi|Q(l-iasigntsi)]r(ds).
•/s2

5.2
ANALYTIC REPRESENTATIONS
239
Since u(t, r) —+ 0 as r —* 0, we have
= lim exp{-^,r)}-l Hm ui^r)
i—>o u{t, r) r-»o r
= -lim^
r-.0 r
lim / idts^rszr-ltsil0)^^)
r- °Js2 r
-ialim [ -({tSl+rs2)<a> ~(tSl)<a>)r(ds))
=: —[Li — iaL2}.
To evaluate L2, we express the corresponding integral as a sum of two integrals
Qi(t,r) and Q2(t,r), the first over 52 fl {s : |tsi| > 2|r|} and the second over
52 fl {s : \ts\\ < 2\r\}. The mean value theorem gives
(tsi + rs2)<a> - (ts,)<Q> = as2r u0'1,
where u € (|tsj| A \tsi + rs2\, \ts\\ V \ts\ + rs2\), and so, for any si ^ 0,
the integrand of Q\ converges to S2|tsi|Q_1. The integrand is dominated by an
integrable function because |isi| > 2|r| implies u 6 pf^, 2|t| and therefore
\u\a~l < |tsi/2|a_1 + |2t|Q_1. This is certainly integrable with respect to T if
a > 1. If a < l,on52,
M4*-1 < cic-i(is,ri+,/|S,r,/ +1) < C7i*r-i(i3,|-" +1),
which, by (5.2.4), is integrable with respect to I\ Therefore, by the dominated
convergence theorem,
limQ1(t,r) = Q|tr-1 / ^Is.r-TCds).
»■—o Jsi
Now consider
Q2(t,r)= [ -l(|ts1j<2|r|)((tSl+rS2)<">-(ts1)<Q>)r(ds).
JS2 r
Suppose firstly a < 1. When |tsi| < 2|r| and {sus2) € 52, the integrand is
majorized by
\r\-lC\rs2\a < |rrIC|rnS,ns,r < |r|-1C'|rr+"|t|-1Slr1'
< C\t\-'\al[-',

240
NON-LINEAR REGRESSION
5.2
for, say, \r\ < 1. as a + v — 1 > 0. Since this is integrable with respect to I", we
get lim Qi{t, r) = 0 by the dominated convergence theorem. The case a > 1 is
r->0
similar. Therefore, by (5.2.11),
a \ | + |a — 1
L2 = a\t\a-1 [ s2\sl\a-1r{ds) = aa?\\t\
Turning to L\, we have
L, = at<a~x> f s2s<Q-1>r(ds) = ourfnt
Js,
a.<a-\>
by (5.2.10). We conclude by (5.2.13),
3<£x(t,r)
dr
= e-antreia/3lCTrt<°>QCTa(^<a-.>
ia\\t\a-1). (5.2.14)
-=o
A similar argument applied to (5.2.12) yields
£(X2|X, =*) = -;
&"" {t*MJ
dt. (5.2.15)
2tt/x,(z)
Substituting (5.2.14) in (5.2.15), we obtain
E(X2\Xi = x)
/OO
e-itxeia^ft<->e-0f\tr (Kt<a-l> _ ioA|t|e-l)(ft
-(
iao
a r°°
2irfx,{x).
aa?
nfx, (x)
[K(/l!+Jl2) + aA(/2i+i,22)],
(5.2.16)
where
/•OO
In = / sintacos(a/8iaftQ)e-,7"'aiQ-1<ftl
Jo
/•OO
1,2 = - / costxsin(a/3,aftQ)e-a°t°'tQ-1di,
Jo
/o
/•OO
J21
/ costxcos{apia?ta)e-'T?tata~ldt,
Jo
/•OO
J22 = / sinixsin(a/31o-fiQ,)e_CTf>tQtQ-1rfi.
Jo
After integrating by parts,
x f°°
/,, = -— / e_ffftQ[a/3, sin(a/3,afta) - cos(a/3,oft")] costx di,
-fv Jo

5.2
ANALYTIC REPRESENTATIONS
241
aft - '°°
hi = —jf
1 x
hl = K + K
+ — / e-a"ta [aPi cos(aP{afta) + sin(a/3l0-f ta)\ sin tx dt,
K Jo
+ — / e-"7"*" [a/3, sin(aAcrfta) - cos{o/3i<Tf ta)) sin tx dt,
-K Jo
x r°°
hi = —^ I e'"^" [aft cos(a/3iafta) + sin(a/Vf tQ)] cos tx dt,
where if = aaf (1 + a2/3f). Therefore, by (5.2.6) and (5.2.9),
in + hi = —^77-—T^i-aft + x-nfxAx) + aftxH(x)},
aaf(l + a'-pi)
721+722 = aaf(l +a*ff)[1 +-a^xirfxl(x)-xH{x)].
Substituting these expressions in (5.2.16) yields
F(Y\X ^ *±a%\ a(\-0lK)\ 1 xH(X)
E{X2\Xi=x) = ——5-55-X +
1 + &ft[ 1 + a*0\
a(X — ftn)
KX +
1 + a2/#
aftx +
nfx,(x) 7t/X|(x)
1 - xH{x)
Remarks
1. To understand the reason for the last statement in the theorem, recall that
when a < 1 and ft = 1, the random variable Xi is totally skewed to the
right and when a < 1 and Pi = — 1, X\ is totally skewed to the left. The
density function fx, (x) vanishes for x < 0 when Pi = 1 and it vanishes
for x > 0 when Pi ~ — 1. Therefore, conditioning with respect to Xi = x
makes no sense when x < 0 if Pi = 1 or when x > 0 if Pi = —1. When
either a < 1, /3i ^= ±1 or a > 1, the support of the density fx,(x) is the
whole real line.
2. The normalized covariation [X2,X\}a/a° is called here k. If X is
symmetric a-stable, a ^ 1, we have T symmetric, Pi = 0 and A = 0 and
hence,
E(X2\Xi = x) = kx for a.e. x.
We thus recover the statement of Theorem 4.1.2 in the case 1 < a < 2.
3. Observe that
H(x) _lm J™ e~itx<j>Xl(t)dt
TrfXl(x) ~ RsJ^° e-^(j>Xl(t)dt'

242
NON-LINEAR REGRESSION
5.2
4. As can easily be seen from the proof of the theorem, the following expression
is equivalent to (5.2.8):
E(X2\Xi=x)
„ „ „ x C e-^"^-1 cos(xt - af31(rftQ)dt
= KX + aa?a(\-plK)Jo ^a(Q -i / J' •
J0 e CTil cos(xt - apxcrfta)dt
(5.2.17)
5. The constants k and A in the theorem are finite when a < 1 because, by
(5.2.4),
< oo
|/4,r'rW</N|SlrH^</^
\Js2 Js2 N" ^s2 l«ir
since |si| < 1, \s2\ < 1 and a — 1 + u > 0.
6. If ATi is (marginally) symmetric, then /3i = 0 and
g(X2lXi =x) = /tx+ftan^>)A , ..(1-j / e-0'"''* sintxdt ]
V 2/ 7r/X|(z) V Jo J
We now turn to the case a = 1.
Theorem 5.2.3 (case a = 1). Lef (Xj, X2) foe a-stable with a = 1 and spectral
representation (T, 0) satisfying Condition 5.2.J. Then, for almost every x,
E(X2\Xi = x) = fco + k(z - Mi) H o (a: - Mi) - ffi
7T Pi
U(x)
(5.2.18)
jB(X2|X, = x) = -^fc0 + k(x - mi) - — A-^r
t 7r 7r/x,(a;)
1//3, = 0. Here,
U(x) = J e-a,tsm(t(x-p.l) + -Pi<Jit\nt\dt,
r°°
V(x) - I e-<7,t(l+lnt)(cosi(a;-/xi))dt,
Jo
ko = — / 52ln|si|r(ds),
K = I^Zlli = 1 f S2S<o>T(ds) = — f s2 sign (a,) r(ds),
/" s2r(ds)
Js,
(5.2.19)
(5.2.20)
(5.2.21)
A =
o\ Js2

5.2
ANALYTIC REPRESENTATIONS
243
and
Hi = — / siln|si|r(ds).
(5.2.22)
Proof: We have, as in (5.2.13),
d4>x(t,r)
dr
^x(t,0)limIf^M-l
*=o
'—or V^x(*.0)
=: -<f>x{t,0)
L\ +i-L2
7T
(5.2.23)
where
Lx = lim / -[|tsi+rs2|-|tsi|]r(ds),
L2 = lim / ■-{(tsi+rs2)ln|tsi+rs2| — tS|ln|tsi|]r(ds).
Clearly,
Lx = sign t I s2 sign (si)r(ds) = aiKt<0>. (5.2.24)
As in the case a ^ 1, to evaluate L2, we write L2 = lim(<2i(£,r) + Q2(t,r))
i—>0
where Q\ involves integration over S2 n {s : |tsj | > 2|r|} and Q2 over S2 D {s :
|isi| < 2|r|}, and we apply the dominated convergence theorem.
We give details only for Q2. Let / : [0, oo) —> [0, co) be defined by f(r) =
r|lnr|, /(0) = 0. For \r\ small enough (0 < \r\ < e_1), / is monotone
increasing and therefore, when \ts\\ < 2\r\ and 0 < \r\ < (3e)_1, one has
|r|-'(/(|fe, +rs2\) + /(|ts,I)) < |r|-'(/(3|r|) + /(2|r|)) < 2|r|-'/(3|r|)
<6|ln3|r||<6|lnl/(||tsi|)| < 6(|ln(§|t|)| + |s,|-"/2), which is integrable by
(5.2.4). Applying the dominated convergence theorem, we have lim Q2{t, r) = 0.
r—>0
Therefore,
L2 = lim Q\(t,r) + lim Q2(t,r)
i—>0 r—>0
L
s2(l+ln|tsi|)r(ds)
= (l+ln|t|) / s2r(ds)+ f s2ln|s,|r(ds)
JSi JS2
= ffiA(l+ln|i|)+tr,fco. (5.2.25)
Substituting (5.2.24) and (5.2.25) in (5.2.23) yields
d<t>x{t,r)
dr
= -4>x(t,0)ax
-=o
Kt<a>+ilX(\ + \n\t\) + i-k0
IT ft

244
NON-LINEAR REGRESSION
5.2
The dominated convergence theorem applied to (5.2.12), with fi\ as in (5.2.22),
gives
dt
2*fX,{x)E(X2\Xl = x) - -ij^ (^f^LJ
/oo
-oo
where
-»t(x-Mi)e-»i/3|ffiiln|t|e-<7i|t|
Kt<0>+i-A(l+ln|t|) + i-Ak)
7T 7T
-5- kE/(x) A(72i + hi) - -konfXx{x)
2. L 7T 7T
(5.2.26)
J2j = / e""ff,t cos t(x-^i)(l +ln£)cosf-/31critlniW
722 = -/ e_CT|'sini(a;-/ii) (1 + lnt)sinf-/3i<7ir.ln*W
(i) Case fi\ ^ 0: After integrating by parts,
J2I = —- / e-CT|tcosi(x-/ii)sin(-/3icritln£W
2pi Jo V7r '
-—(x — fi\) / e_<T,tsini(x — /ii)sin( -/3\ait\nt)dt,
1\(T\ J0 \7T /
W
122
so that
—- / e *'* sin£(x - /ii)cos( — f3\0\tlnt)dt
•Pi Jo V7r '
/•OO f\
-—(x-/xi) / e_<T|tcosi(x — /ii)cos( -0\o\t\nt]dt,
20:
hi + hi
■——(x - M,)7r/X,(x) +-^-[/(x).
2pi<Ti 2pi
Substituting this expression in (5.2.26) and rearranging the terms, we obtain
E{X2\XX = x)
7T Pi Pi KJXAX)
la
A-/3.K r, . U{x) 1
7T Pi L Trfx,{x)l
k0 + k(x —/ii) +
(ii) Case (3\ = 0: After integrating by parts,
U(x) = r e-CTlt sint(x - m)dt = nfx^x\x - Mi).
Jo a\

5.2
ANALYTIC REPRESENTATIONS
245
Since ^
h\+In= / e-£r,tcosi(x-/x1)(l + lnt)dt = V(x),
Jo
Relation (5.2.26) becomes
E(X2\Xl = x)
o-\
tf/x, (?)
K^l*M{x _ Ml) _ lXV(x) - -*Wx,(x)
Cl 7T 7T
= fco + k(x — fj,\) A-
7T -K TT/x, (x)
Remarks
1. The shift parameter \i\ also appears in expression (5.2.7) of /x, (x).
2. The constants re and A are defined as in Theorem 5.2.2. When a = 1, the
constant A is proportional to the skewness parameter of X2-
3. If/3,^0,
U{x) ^lmj^e-itx4>x,(t)dt
tt/x,(x) Re/^c-^^x.Wcft'
The following corollary shows that the regression is linear when X\ is totally
skewed to the right ((5\ = 1) or when it is totally skewed to the left (fix = —1).
Corollary 5.2.4 (case fix = ±1). Suppose that the conditions of Theorems 5.2.2
or 5.2.3 hold. If fix — ±1, then, for almost every x,
{rex, if a 7^ 1,
(5.2.27)
— ^-fco — re/ii + rex, if a = 1.
In the case a < 1, this relation is well defined only for x > 0 when fix = 1, and
for x < 0 when fix = —1.
PROOF: fix = 1 implies si > 0, T-a.e., by (5.2.1) and (5.2.2) and therefore
A = re by (5.2.10) and (5.2.11). Similarly, fix = -1 implies s\ < 0, T-a.e. and
re = -A. In both cases, A — p\re = 0, and the corollary follows from Theorems
5.2.2 and 5.2.3. I
The next result shows that the regression E(X2\Xx = x) is typically
asymptotically linear as x -» ±oo. We know from Corollary 5.2.4 that the regression is
linear when fix = ± 1. For other values of fix, one has:

246
NON-LINEAR REGRESSION
5.2
Corollary 5-2.5 (Asymptotic relations). Let (X\,X2) be a-stable, 0 < a < 2,
with spectral representation (T, 0) and suppose that Condition 5.2.1 holds if
0< a< 1. Then, for Pi ^ ±1,
K + X
E(X2\Xl = x) ~ ——x , x -► oo, (5.2.28)
i + P\
and
k — X
E(X2\XX = x) ~ —x , x — -co. (5.2.29)
1 - p\
Proof: It is sufficient to consider the case x —♦ oo. Indeed, consider the vector
(-X\,X2) whose parameters are f3\ = -/?,, k = -k and A = A. Since
E(X2\X\ — x) = E(X2\ — X\ = -x), one can obtain the asymptotic behavior
of E{X2\X\ = x) as x -> -oo from that of .Ep^l^i = x) as x —► oo by
replacing /3i by -/?i, «; by —k, and x by -x = |x|. Suppose f3\ ^ ±1.
We study first the cases with a ^ 1 and then those with a = 1.
(a) case a^l, x -+ oo: To obtain the asymptotic behavior of E(X2\X\ = x),
we use its expression in terms of Uj, i,j = 1,2, as given in (5.2.16). By
(Titchmarsh 1986, Theorem 126), as x —> oo,
/,,~r(a)(sin^)x-Q, In = 0(x~a),
(5.2.30)
72, ~ r(a)(cos ^)x-Q, I22 = o(x-Q),
when 0 < a < 1 and also when 1 < a < 2. (If 1 < a < 2, use integration by
parts to transform the terms tQ_1 in Iij, i, j = 1,2, into ta~2.) Moreover,
fXl(x) ~ i(l +/31)(sin™)a?r(a+ l)*"0-1
(Ibragimov & Linnik 1971, Theorem 2.4.2). Substituting in (5.2.16) and using
a = tan 3p, we obtain
£[X2|Xt=x] ~ c(Kr(a)(sin^)x-Q + aAr(a)(cos^)x-Q)
k +A
where
1+/3:
aor
x,
■k (l+A)(sin^Kr(l+a)'
(b) case a = 1, f3\ ^ 0, x —► oo: Since x — p,\ ~ x as x —* oo, we may assume
/ii = 0. Then,
l/(x) = / e~CT|' sin (tx +-A cti t In i)di = l/,(x) + ^(x),

5.2
ANALYTIC REPRESENTATIONS
247
where
dt,
Ui(x) = / e^'sinZ-^tlntjcostx
U2(x) = / e~"1' cos (-Picrxtlnt) sin tx dt.
After integrating by parts,
Ui(x)
1 r°° r /2 \
= — / e~CT,t -o^sin -cTi/Mlnt)
+—cri/3i(l +lni)cos(—(Ti^itlntj sin txdi,
1 J /"» r /2 \
I72(x) = - + - / e~CT|t -^cosf-fTiAHnf)
+—Cij0i(l + lnt)sin( — (TiPitlnt J costx dt.
Since the factors of sin tx and cos tx are integrable, we see that U\ (x) =
o(x-1) and f/2(x) ~ x_1 as x —> oo by the Riemann-Lebesgue lemma (Titch-
marsh (1986), Theorem 1). Therefore,
U(x)
1
Since, as is well known,
we obtain
/x,(z) x ,
(5.2.31)
(5.2.32)
f/(x)
1
7T/X,(X) tXi{l+Pi)
which, substituted in (5.2.18), gives
A-/3i«
x,
E{X2\Xi = x) ~ kx +
A
x —
l+0i
K + A
i+a
■X.
This is the same result as in the case q^I.
(c) case a = 1, 0\ = 0, x —> oo: If Y is a SaS random variable with
q = 1 and scale parameter a\, then by the inversion formula (Lukacs (1970),
Theorem 3.2.1), P(0 < Y < x) = £ /0°°e'-<T|*t-1 sinix dt. On the other

248
NON-LINEAR REGRESSION
5.2
hand, because of the symmetry, lim^oo P(0 < Y < oo) = 1/2. Therefore,
limx^oo J"0°° e~a,tt~l sin tx dt = \ and hence
/•OO
V(x) = / e-a,t(l+\nt)costxdt
Jo
/-OO i /"OO
= — / e_CT|t(l+lnt)sintedi-- / e~a[trx sinix dt
x Jo x J0
7T 1
~ -x-, (5-2.33)
2 z
since the first integral is o( 1) by the Riemann-Lebesgue lemma. Substituting
(5.2.33) and (5.2.32) in (5.2.19) yields E(X2\Xi = x) ~ (/t + A)z, i.e., (5.2.28)
with /3i = 0.
This completes the proof of the corollary. I
Remark. If k ± A ^ 0, then the regression is asymptotically linear. If either
k + X = 0otk — A = 0, then the respective tail of the regression does not grow
in absolute value as fast as \x\.
The following corollary gives a necessary and sufficient condition for linearity
of the regression.
Corollary 5.2.6 (Linearity). Let (Xu X2) be a-stable, 0 < a < 2, with spectral
representation (T, 0) and suppose that Condition 5.2.1 holds if 0 < a < 1. Then
the regression E(X2\X\ = x) is linear if and only if
A = /?,«. (5.2.34)
//A = 0{k, then E{X2\X\ = x) is given by (5.2.27) for a.e. x.
Proof: If A = 0\n, then E(X2\X\ = x) = kx for a.e. x by Theorem 5.2.2
when a^l, and E(X2\X\ = x) = -^fco + k(x - fx\) for a.e. x by Theorem
5.2.3 when a = 1.
Suppose, now, E{X2\X\ = x) = Ax + B for a.e. x and also, to the contrary,
that X^= P\K. In view of Corollary 5.2.4, this contradicts either (5.2.8) or (5.2.18)
if /3i = ±1. If 0i ± ±1, then, by linearity,
.. E{X2\X, = x) E{X2\X> = x)
urn = lim .
i—>oo X x—> — oo X
In view of (5.2.28) and (5.2.29), this implies
K + X K — X
1+/3, = \-0x

5.2
ANALYTIC REPRESENTATIONS
249
and hence A = (5\ k, a contradiction. I
In particular, in the SaS case, we obtain the following extension of Theorem
4.1.2:
Corollary 5.2.7 // {X\, X2) is SaS, 0 < a < 2, with spectral measure T and if
Condition 5.2.1 holds when 0 < a < 1, then
E(X2\X\ = x) = Kxfor a.e. x.
Example 5.2.8 The regression can be linear even though (X\, X2) is not
symmetric. For example, suppose a > 1 and let X = (Xi, X2) have spectral measure
^-s((7!'^))+i((-^-7!))+i«<,'-1)).
where 6((x\, x2)) denotes a unit mass at the point (x\, x2). The vector X is not
symmetric because T is not symmetric. However, f3\ = 0 (X\ is SaS) and A = 0.
Since (5.2.34) is satisfied in this case, the regression is linear. The slope is k — 1
because both ct" and the covariation fs_ S2sfa~l>T(ds) equal 2(-4=)a.
Example 5.2.9 The regression may be non-linear even though both components
X\ and X2 are SaS. For example, let a > 1 and X = (X\, X2) have spectral
measure
r = 6((lI0)) + 5((0,l))+2«/25((-^>--L)).
In this case, X is not symmetric, X\ and X2 are SaS, and the regression is not
linear because /?, = 0 but of A = /& s?|s, |Qr(ds) = -2Q/2(^=)Q+I ^ 0.
Integral Representation
In applications, (^!, X2) is often given by its integral representation
(XUX2) = (7 /,(x)M(dx), J f2{x)M(dx)\ , (5.2.35)
where M is an a-stable random measure on the measure space {E,£) with
control measure m and skewness intensity (3{-) : E —> [—1,1]. The following
proposition expresses Condition (5.2.4) and the constants of the regression in
terms of f\, f2, /?(•) and m(dx).
Proposition 5.2.10
rm<00* [ i^r(ds)= / \i^im(dx)<00,
(5.2.36)

250 NON-LINEAR REGRESSION 5.2
where E+ = {x 6 E : /,2(x) + /22(x) ^ 0}. Moreover,
°x = f \Mx)\Qm(dx),
JE
A = -4 / fdx)<a>P(x)m(dx),
°~l JE
M. = ~/ /i(»)(ln '^glf=)^(x)m(dx)
)f3(x)m(dx)
fco =
k =
1 /" frr^ln l/l(a:)l
vJEj2{x)yn7m^m
[X2,Xi)a
(case a = 1),
(case a = 1),
-L f f2{x)h{x)<a-l>m{dx),
°\ Je+
X = -1 / /2(x)|/,(x)r-^(x)m(dx).
Proof: Suppose firstly a ^ 1, and set a = tan ^ and |f | = x//? + /f. The
characteristic function of (Xi, X2) is
2 .
Eexp/i^^ / /j-M(dx)}
2 2
2
2 2
= exp{/ he*H°+ia(S^djSj)<a>]nds)}
by making the change of variables
T : E+ —» 52, T : a: h-> (si, S2) =
and setting T(ds) = r+(cfe) + T_(—ds), with
/i(s) f2(x)
|f(x)|'|f(x)|
r±(ds) = 1±^X\{(x)\am{dx), x = T-'(s).

5.3
EXAMPLES
251
Hence, in order to transform an integral involving s\, s2 and r into one involving
/;, f2 and m, one expresses it as a sum of two terms: the first is obtained
by replacing Sj by $g|, j = 1,2, and r(ds) by ±(1 + 0(x))\f{x)\am(dx),
and the second is obtained by replacing Sj by ~Jm? , j = 1,2, and r(ds) by
i(l - P(x))\f(x)\am(dx). (The same rule applies when a = 1.) For example,
when q = 1,
fco = / sj ln|s2|r(rfs)
Js2
h+mrwj—r-|f|m(dx)
jf /,(lnJj|i)/3(x)m(dx).
Similarly,
/.
521 r(d8)
S2 lsl
=L -mrm{dx)- ■
5.3 Examples
Here, we apply the results proved in the preceding sections to various important
classes of a-stable laws, namely those of moving averages, sub-Gaussian and
harmonizable vectors.
Example 5.3.1 Moving averages
Often, (Xi,X2) = (X(U),X{t2)) where t, < t2 and {X(t),t £ T} is
some a-stable stochastic process. If the process X(t) has a representation of
the type X(t) = J^ f(t - x)M(dx) and M has Lebesgue control measure,
then there is no v > 0 satisfying Condition (5.1.27) and, clearly, if 0 < a < 1,
E(\X{t2)\\X{ti) = x) = oo. This is the case, for example, ifX(t) = /„' M{dx)
(a-stable Livy motion) or if X(t) = ji^ e_A<1_x)M(cte) with A > 0 (Ornstein-
Uhlenbeck process).

252
NON-LINEAR REGRESSION
5.3
Relation (5.2.36) however, can be satisfied by two-sided moving
averages. Consider, for example, the two-sided a-stable Omstein-Uhlenbeck process
■^(*) = I^°oo e~A|t-x'M(ota) where M has Lebesgue control measure. Then any
v > 0 satisfies (5.1.27) and, therefore,
(0 E{\X(t2)\p\X{ti) = x) <ooa.e. for any p < 2a + 1.
In particular, E(X2(t2)\X{t\) = x) < oo a.e. if a > 1/2.
(//) If X is SaS,
E(X(t2)\X(U) = x)
= ( -Xa / exp{-A(|t2 -u\ + (a- \)\t\ - u\)}du j x a.e.
Example 5.3.2 Sub-Gaussian vectors
Let 0 < a < 2, let G\,G2 be zero mean jointly normal random
variables and let A be a positive a/2-stable random variable, independent of
(G{,G2) with Laplace transform Ee~iA = e-<a,\ 7 > 0. Then (XUX2) =
(yl'^Gi, -A1'/2G2) is a sub-Gaussian SaS random vector with underlying
Gaussian vector G = (G\,G2) (see Sections 2.5 and 3.7).
Corollary 5.3.3 Let X = (X\,X2) be a sub-Gaussian a-stable random vector
with underlying Gaussian vector (Gj, G2). Then
(0 E(\X2\P\XX = x) < 00 a.e. for anyp<a+l.
In particular, -Epffl.X'i = x) < 00 a.e., // 1 < a < 2.
(//) //X is SaS,
E(X2\X>=x) = C°y°^2)xa,. (5.3.1)
The proof uses the following lemma which will also be useful in the sequel.
Lemma 5.3.4 Let X = (X\,X2) be a sub-Gaussian SaS random vector with
underlying Gaussian vector G having covariance matrix
1 r
, ,, l<r<l.
r 1
Let TT denote the spectral measure of the sub-Gaussian SaS vector X. Then for
any 0 < v < 1, there is a finite positive constant K independent ofr such that
L
121 -rP(ds) < K. (5.3.2)
s2 isil

5.3
EXAMPLES
253
PROOF: By Proposition 3.7.1,
(X,, X2) i C (f GMM[duj), J G2{w)M{dw)\ , (5.3.3)
where (Q, T, P) is the probability space on which the Gaussian vector {G\, G2)
is defined, M is a SaS random measure on (Q, T) with control measure P and
C = G(q) is a finite positive constant. Using (5.1.27), we obtain
Since (Gi,G2) = (G1,rG1 + (l-r2)'/2e) whereeisan7Y(0, l)randomvariable
independent of G\, we have
^ |gV~ = g IG.I" 2Q+w(^iG!i|Q+^er+,'S|G1|-'))
proving the lemma. I
PROOF of Corollary 5.3.3: We assume, without loss of generality, that EG] =
EG\ = 1 and use the notation of the preceding lemma. That lemma, Relation
(5.2.36) and Theorem 5.1.3 prove part (i). Corollary 5.2.7 shows that E(X2\Xi =
x) = kx a.e. To identify k, we use the representation (5.3.3) to obtain
f |s,|Qrr(ds) = Ca f \Gi(u>)\aP{du) = CaE\G,\a (5.3.5)
Js2 J a
and
/ s2sfa-l>rr(ds) = Ca f G2(w)Gi(o;)<Q-1>P(dw)
is2 Jn
= CaEG2Gfa~l>
= CaE{rGi + Vi-r2e)G<a-l>
= CarE G, Gfa~l>
= CarE\Gi\a, (5.3.6)
which gives the covariation . Hence, k = r when a] := EG] and a\ := EG\ are
both equal to 1. In the general case,
\a2 ai a\ J a\
Xi x\ a2r Cov(Gi,G2)
1 = —x = —^—^ x a.e.
VarGi
Statement (i) in Corollary 5.3.3 can be verified directly as follows. Let p <
a + 1 with 0 < a < 2 and write G2 - t + aG\ where e is Gaussian, EeG\ — 0
and a is a constant. Then
EflA'^l'I^G, = s) = Efl^e + ax\*\Al'2G{ = x) < cc

254
NON-LINEAR REGRESSION
5.3
because a simple computation shows that the conditional density g(y\x) of A^2e
given AX/1G\ = x satisfies g(y\x) ~ C(x)y~2~a as y —> oo, C(x) > 0. (Use
Exercise 5.1.)
Example 5.3.5 Real harmonizable vectors
Let M be a (complex-valued) rotationally invariant SaS random measure on
(K, B) with & finite control measure m, let i,, t2 be real numbers and let
/oo
e^xM(dx), j = 1,2.
-oo
Then {X\, X2) is a real harmonizable SaS random vector. (See Example 3.6.7.)
Corollary 5.3.6 Let X = {X\, X2) be a real harmonizable SaS random vector
as above. Then
(0 E{\X2\P\XX - x) < 00 a.e. for any p < a+ 1.
In particular, E{X2\X\ = x) < 00 a.e. if\<a<2.
(ii) IfX is SaS,
E(X2\Xi = x) = ( J m(dy)yj (f cos((t2 - t,)l/)m(dy) J x a.e.
Proof: . By (3.6.6), the joint characteristic function </>x(#i, 02) of the SaS
random vector X = (X\, X2) has the form
£7expi(fliXi+ff2JC2) = exp{-6i(a) /" |02 + 6»§ + 26^6>2CosAy|Q/2m(dy)},
(5.3.7)
where b\(a) is a finite positive constant and A = t2 — t\. By Proposition 2.5.2,
for any y € K,
[6>f + 6>2 + 261,612cosAy|a/2 = ^(a) / 161,5! + fl2s2rrCOiAl/(ds), (5.3.8)
where 62(a) is a finite positive constant and where we use the same notation as in
Lemma 5.3.4. Therefore
<hc(eu02) =exp{-bi(a)bi(a) J \9\SX+62s2\a J rcosAy(ds)m(dy)}.
Comparing this expression with <j)y.{8\,62) — exp{—/s \Q\S\ + 02S2\°T(ds)}
and using the uniqueness of the spectral measure, we conclude that the
spectral measure T of the real harmonizable SaS random vector (Xi,X2) can be

5.4
GRAPHICAL REPRESENTATIONS
255
represented in the form
/oo
r„&y(A)m(dy) (5.3.9)
■oo
for any Borel set A on 52.
For any 0 < u < 1, we have, using Lemma 5.3.4,
/ -jhrT(ds) = *>i(aMa) / / ij—-rcosAv(ds) m(dy)
JS2 \s\\ J-oa Us2 \SU
/oo
Km(dy) < oo.
•oo
Theorem 5.1.3 implies now that for almost every x, E(\X2\P\X\ = x) < oo for
any p < a + 1. To prove (ii), we use Corollary 5.2.7 and (5.3.9). Computing the
covariation, we obtain
f s2sfa-]>T(ds) = b{{a)bi{a) f [ a28<a-l>rcmAy(ds)]m(dy)
JS2 J-oo iJs2
/oo
cos Ay m{dy)
-oo
by (5.3.6). Moreover,
f \sl\aT{ds) = bx(a)b2{a)CaE\Gx\a f°° m(dy),
J S2 J—OO
by (5.3.5). This establishes (ii). I
5.4 Graphical representations
In this section, we present graphical representations of the regression functions
given analytically in Theorems 5.2.2 and 5.2.3. They were obtained by using
the software package for computing bivariate regressions and probability density
function of a-stable random variables given in Hardin, Samorodnitsky and Taqqu
(1991b). That paper includes a description of the package and the listing of the
source code, written in the C language.
In the case a ^ 1, the five parameters a, /?i,«, A, and a\ given in Proposition
5.2.10 are required to describe the regression function completely. In the case
a = 1, these five parameters, together with fco and fi\, are required to describe
the regression function completely. Thus, there is a myriad of possible choices
for these parameters which are consistent with their definitions, giving rise to an
unmanageably large family of regression functions. In producing the plots that

256
NON-LINEAR REGRESSION
5.4
follow, we have chosen to restrict the parameter space considerably, but in a way
which we hope will give the reader some feeling for the general character of these
functions.
To accomplish this, we use the stochastic integral representation (5.2.35) for
(X\, Xi). Here, the random measure M is taken to be totally right-skewed (i.e.,
P(-) = 1), with domain E = [0,1] and Lebesgue control measure m. The
functions /_,-, illustrated in Figure 5.3, are restricted to those of the form
/i(*) = l[0,ci)(*)-l(ci,l](t),
where 1,4 is the indicator function of the set A. With these restrictions, all
scale parameters equal unity and the three parameters a, c\ and C2 determine the
regression function.
JjM 0
Figure 5.3: The function fj(t).
Regressions for (X\, Xj) in this class may be interpreted as regressions involving
three independent stable variables as follows. Define Cm\n = min(c], C2) and
c„,ax = max(ci, C2), and let Z\,Z2, and Z?, be independent identically distributed
totally right-skewed (/3 = 1) a-stable random variables with unit scale parameter.
Then (X\, X2) is distributed as (Fi, 5^). where
Yi = cJ^Z, ± (-l)''+'(<w - cmin)l/«Z2 - (1 - cn*)""^

5.4 GRAPHICAL REPRESENTATIONS 257
and the upper sign holds if Cm^ = c\, and the lower sign prevails otherwise. For
example, if ci = 0.5 and c2 = 1, then the regression of X2 on X\ is the regression
of k{Z\ + Z-i) on k{Z\ — Z-i) for the appropriate constant k.
From Proposition 5.2.10, the parameters defining the regressions in Theorem
5.2.3 are
<ri = 1,
j9, = 2c,-1.
Hi = -In(V2)/?,,
7T
fco = - ln(V2)A,
k = l-2|ci-e2|,
and
A = 2c2-1.
We shall refer to the skewness parameter for Xi as fa. For the present class of
distributions, /% is equal to A.
The asymptotic results of Corollary 5.2.5 are illustrated in the graphs and
translate to the present parametrization as follows:
lim g(*2|Xi = s) f 1 if02>c,,
x^+oo X \ 2(C2/Ci)-1 ifCj>C2
and
Hm E{X2\Xi = x) __ f 1 - 2(c2 - c,)/(l - cQ ifC2>c,,
x-»—oo X 1 1 if C] > Ci.
The limits here do not depend on the value of a. Moreover, by Corollary 5.2.6,
the regression is linear when c\ = c2 (here, this corresponds to X\ = X2).
The regression functions are computed by making use of Theorems 5.2.2 and
5.2.3. Unfortunately, the functions H(x),U{x),V(x), and the density function
fx, (x) do not in general have representations in terms of elementary functions, and
thus their values must be computed by means of numerical integration. Although
the general problem of computing the integrals is straightforward in principle, the
specific task is fraught with difficulties. The next section discusses the numerical
techniques in detail.
In the graphs that follow, two of the parameters a, c\ and c2 are held constant
while the third varies. Although not all possible variations are illustrated, much of
the behavior for parameter choices not illustrated can be inferred from the graphs.
Figure 5.4 corresponds to c\ = 0.5 and c2 = 1 and, hence, to 0i = 0 and
02 = A = 1. It shows the regression of a totally right-skewed variable Xz

258 NON-LINEAR REGRESSION 5.4
10
X
II
-5
-10
-10
0
x
10
Figure 5.4: Regression functions for /?i = 0, ft = 1, a varying.
upon a symmetric variable X\ for selected a, or, equivalently, the regression of
k(Z\ + Z-i) on k(Z\ — Z2) as mentioned above. When a ^ 1, the value of these
regression functions at the origin has the same sign as a = tan ^ and hence is
positive for a < 1 and negative for a > 1.
Figure 5.5 corresponds to cz ~ 0.5 and various values of c\. It represents
a regression of a symmetric variable Xi upon variables X\ with skewness /3i
varying from 0 to 1, for the case a = 1.9. The value ci = 0.5 corresponds to X\
symmetric, in which case X\ = X2 and tiie regression is linear. When c\ — 1, X\
is totally right-skewed and the regression is linear at zero.
In Figure 5.6, X2 of varying skewness is regressed on a symmetric X\ for the
value a = 1.5. Here, c\ — 0.5. When cj = 0.5, the regression is linear since
X2 — X\. The curious non-zero intersection of the regression lines occurs for all
fixed values of a > 1 at an x value depending on a but does not occur for values
of a < 1 (see Figure 5.12). Figure 5.7 corresponds also to a = 1.5, but in this
case it is X2 that is symmetric (fh — 0).
In Figures 5.8 and 5.9, a equals 1.1. Figure 5.8 should be compared to Figure
5.7 because they both illustrate the regression of a symmetric random variable X2
upon random variables X\ of varying skewness. Figure 5.9 shows the regression
of a totally right-skewed X2 upon X\ of varying skewness. Here 02 — 1, and

5.4 GRAPHICAL REPRESENTATIONS 259
10
>-N 5
H
II
i—*
X o
-10
-10 -5 0 5 10
x
Figure 5.5: Regression functions for a — 1.9, /3| = 2ci — 1 and ft = 0.
hence ft = A = 1. As the skewness of X\ approaches that of X2, the regression
function approaches the identity, yet the left asymptote always has slope —1.
In Figure 5.10, a — 1. The parameter c\ is chosen to be 0.9, so that X\ has
skewness 0.8. The skewness of X2 varies from —0.8 to 1. (For X2 of skewness
— 1, the regression is the negative of that for X2 of skewness 1.) The value
c2 = 0.9 corresponds to X2 = X\, in which case the regression is linear. This
graph shows that a small change in skewness can result in a large change in the
global shape of the regression function.
Figure 5.11 represents the regressions of variables of varying skewness upon
a symmetric variable for the value a = 0.9. The value c2 = 0.5 corresponds to
X2 = X\, in which case the regression is linear. This plot should be compared
with the case a = 1.5 illustrated in Figure 5.6.
Figures 5.12 and 5.13 both correspond to a = 0.5, but ft = 0 in Figure 5.12,
whereas ft = 0 in Figure 5.13. Observe that Figures 5.6 and 5.12 where ft = 0
have roughly the same shape. So do Figures 5.7, 5.8 and 5.13, where ft = 0. In
Figure 5.13 the regression for ft = 1 (ci = 1) is defined only for x > 0, because
the density of Xx has support on [0,00]. It is linear with slope « = 0.

260
NON-LINEAR REGRESSION
5.5
10
X
II
X
$
KJ
0
-5
-10
\S=1'0
!^=oV.
■c = 0.8\\. y,
■c2=°-7^^x
<^0*/
/c. = 0.5
i i i i ... -
s^%?
^S
-
■
i . . . .
-10
0
x
10
Figure 5.6: Regression functions for a = 1.5, p\ — 0 and (h = 2cj — 1.
5.5 Numerical techniques
We shall work with the integral representation since it is the one which is the most
commonly used in practice. For convenience, we start by summarizing the main
results of Sections 5.1 and 5.2.
The regression formulas
Let
(XUX2) = [J fi(x)M(dx), J h{x)M(dx)) (5.5.1)
be a-stable, 0 < a < 2. A sufficient condition for the regression E(Xj\X\ = x)
to be defined when 0 < a < 1 is
/.
\fi(x)\
a+is
-m(dx) < oo,
for some v > 1 - a, where E+ = {x 6 E : ff{x) + /|(x) ^ 0}.
The following quantities enter in the expression of the regression:
"f = / \Mx)\am{dx),
J E

5.5
NUMERICAL TECHNIQUES
261
10
II
C3
-5
-10
-10
0
X
10
Figure 5.7: Regression functions for a = 1.5, /3| = 2ci — 1 and 02 = 0.
Pi = 4r / /i(x)<a>/3(x)m(dx)I
Mi
fc0
[X2,X,]„ 1
m(dx),
^ = -^/ /2(x)/,(x)<Q-'>
1 CT1 JB+
kJe+ v VfiW + fZW
1 / /2(s)(ln /f2\f\{x)[„. >(*)m(dx)
(case a = 1),
(case a = 1).
ai is the scale parameter of X\, 0i is the skewness parameter of X\, k is the
normalized covariation of X2 on X\, X is the normalized skewed covariation of
X2 on X\, and pi \ and fc0 are shift parameters that appear in the case a = 1. When
/?(■) = 0, the regression is linear j5(X2|Xi = x) = kx with slope equal to k,
the normalized covariation. When /3(-) ^ 0, the formulas for the regression are
different in the cases a ^ 1 and a = 1. In the case q^I,
E(X2\Xl=x)=Kx + a(tan™){\-l3lK)1£^a1, (5.5.2)

262
NON-LINEAR REGRESSION
5.5
10
0.5
X
II
i—f
X
63
-10
-10
-5
0
x
10
Figure 5.8: Regression functions for a = 1.1,0i = 2ci — 1 and P2 — O.
where
r(x) = / e-'V"1 cos(xt - (tan ^)/?,tQ)<ft, (5.5.3)
s(x)= f e-tacos(xt- (tan^)/?,tQW (5.5.4)
(See (5.2.17).) The density function of Xi equals ^-s( f-).
1 „/1 ■
When a = 1 and /3, ^ 0,
2a,. . , , . \-PiKt. , t/(x)
S(X2|X1=x) = -^^ + K(x-Ml) + ^p((x-/x1)-a1^)I
(5.5.5)
where
U{x)= J e-'"tsm(t(x-nl) + -l3icrltlnt)dt, (5.5.6)
W(x)= f e-c"tcos(t{x-n\) + -0ia\t\nt}dt. (5.5.7)
When a = 1 and /3, = 0,
£(X2|Xi = x) = -fco + «(z - Mi)
2o\_xy{x)
TV
n W(x)'
(5.5.8)

5.5
NUMERICAL TECHNIQUES
263
10
-10
-10
0
X
10
Figure 5.9: Regression functions for a = 1.1, j3\ = 2ct — 1 and ft = 1.
where
yoo
V{x)= e-ait{\+\nt){cost{x-n{))dt, (5.5.9)
Jo
and W[x) is as above. When a = 1, the density of X\ equals ^W(x).
Suppose 0 < q < 2. The regression is linear when Pi = ±1, and more
generally, when A = /3i«. It is then equal to kx if a ^ 1 andto/tx — (^ko + Kpi)
ifa= 1.
The regression is typically asymptotically linear. When /3j ^ ±1,
E(X2\XX = x) ~ f^-a:, x
1+/3,
oo,
and
E(X2\Xi = i)
1-/3:
■x, a; —» —oo.
Integrand characteristics
In the given form, the integrals denning r(x),s(x),U(x),V(x) and W(x)
are improper integrals, and all have integrands which decay exponentially while
oscillating, in many cases increasingly rapidly, about zero. The integrand of V
and the integrand of r in the case a < 1 have singularities at zero; otherwise the
integrands are reasonably well behaved.

264
NON-LINEAR REGRESSION
5.5
10
II
.—I
-10
-10
-5
0
x
c2=0.2
c2 = 0.1
10
Figure 5.10: Regression functions for a = 1.0, /?i = 0.8 and 02 = 2c2 — 1.
It might be expected that with the exponential decay of the integrands, the
improper integrals could be truncated at some adequately high cutoff and subjected
to any one of the standard integration techniques, with acceptable results. One
problem with this is, of course, the singularities. Another problem is that as the
parameter x gets large, the sinusoidal oscillations become more rapid, and each
integral decays to zero, imposing more stringent requirements on the accuracies
of the individual integrals to ensure accuracy of the quotient. For example, the
integral r(x) decays as x~a and s(x) decays as x~(a+l\ (See the proof of
Corollary 5.2.5.) For larger x, then, both more iterations and higher truncation levels
may be required. The parameter a also affects the requirements for truncation
due to the slower decay of the integrand for lower a. To illustrate some of the
behavior mentioned above, Figure 5.14 shows a graph of the integrand for r(x)
for a = 0.5 and a = 1.5 when x = 8.
One can make the integrals more amenable to numerical techniques by making
changes of variables in the integrands. For example, the change of variables s = ta
converts r(x) to
r(x) = - r e~s cos(xsi/Q - (tan^)/3lS)ds. (5.5.10)
This eliminates the singularity in the integrand in (5.5.3) when a < 1. The

5.5
NUMERICAL TECHNIQUES
265
-10 -5 0 5 10
X
Figure 5.11: Regression functions for a = 0.9, /3| = 0 and /32 = 2c2 - 1.
same change of variables in the denominator integral s(x), however, introduces
a singularity at zero for values of a greater than one. To eliminate the need for
truncation, the change of variables s — e~l represents r (x), after some fortuitous
cancellations, as
r(x) = - / cos (x( - In s) 1/q + (tan^)/3iln s\ds. (5.5.11)
The same change of variables in the integrand for s(x) introduces a singularity at
zero for a < 1, and at one for a > 1. Although representation (5.5.11) results
in a bounded integrand, the integrand oscillates increasingly rapidly about zero.
Even though the contribution of the integrand from zero to e is bounded by e,
the oscillations for s slightly greater than e may be rapid enough to require an
unacceptably small step size in the numerical procedure in order to obtain the
necessary accuracy.
In the case a = 1, the logarithms in the integrands do not cause singularities or
discontinuities except in V(x). That singularity may be removed at the expense of
introducing an additional improper integral with the change of variables t = e~s

266 NON-LINEAR REGRESSION 5.5
-10 -5 0 5 10
X
Figure 5.12: Regression functions for a = 0.5, /3t = 0 and ft = 2c2 - 1.
for0<* < 1:
/ e_cr,'(l+lnt)cos(i(x-Mi))dt= / e_<"e""-s(l-s)cos(e-s(a:-/ii))ds.
Jo Jo
(5.5.12)
Numerical procedures
Because of the variation of integrand characteristics over the range of
parameters considered, regardless of the representation of the integral, it is important
for the integration scheme to allow truncation (if used) and step size to vary as
the parameters vary. Bounding arguments can be applied to whatever form of the
integrand one chooses in order to find a cutoff point for which the truncation error
is small. Step size is somewhat more difficult to pick, as the usual error formulas
involve derivatives which may be difficult to evaluate, or may have extremely
high values not indicative of the actual errors.
A standard way of solving the step size selection process is to use one of the
(by now standard) versions of the classical midpoint and trapezoidal integration
formulas which allow the step size to be iteratively decreased. The integration
sample point mesh is successively refined, and the corresponding numerical
approximation is successively refined by updating the previous computations. When

5.5
NUMERICAL TECHNIQUES
267
-10 -5 0 5 10
x
Figure 5.13: Regression functions for a = 0.5, /3| = 2ci - 1 and ft = 0.
successive approximations differ by some small prespecified amount, the
iterations are terminated. Approaches such as these seem suited to the problem at
hand since different parameter choices have different requirements.
Unfortunately, these methods are slow to converge for our problem and tend to require
more computation than other methods. This problem is not alleviated by any
of the changes of variables mentioned. In fact, for some representations, the
approximations did not converge.
Another promising possibility is to use a variable step size formula, since for
all of the representations considered, the integrand varies much more rapidly in
one part of the domain than in other parts. The bulk of the computation could
take place in the regions of greatest change, enabling faster convergence and
more control over the accuracy. Unfortunately, as for the iterative methods, this
seemingly good idea fails the test of a practical implementation.
In the choice of integrand representation and integration technique one must
consider computational expense, trade-offs involving number of iterations,
number and complexity of function evaluations, and numerical accuracy. For example,
in the case a ^ 1, the integrands r(x) and s(x) in (5.5.3) and (5.5.4) are identical
except for the factor ta~l. It is tempting to try an integration scheme which reuses
the function evaluation for s in computing r in order to decrease the computational
overhead. The savings realized in doing this, however, do not outweigh the costs

268
NON-LINEAR REGRESSION
5.5
< a = 0.5 p\= 0.0
a = 1.5
0 1 2 3 4 5 6
Variable of Integration
Figure 5.14: Integrand in (5.5.3) for x - 8.
incurred, compared with other methods, due to a larger number of iterations.
Many combinations of integration technique and integrand representation were
tried. The combination that met with the most success was the version of
Simpson's rule known as the "one-third" rule, coupled with the representation (5.5.10)
for the function r(x), the representation (5.5.4) for the function s{x), the
representations (5.5.6) and (5.5.7) for U(x) and W(x), and the representation for V(x),
obtained by writing the integral in (5.5.9) as the sum of two integrals, one from 0
to 1 and the other from 1 to oo, and using the representation (5.5.12) for the first
of these. The remainder of this section describes this approach in more detail.
To choose the step sizes, the following approach is taken. Require a certain
minimum step size, but allow it to be chosen smaller, based on the frequency of
the sinusoidal term. This frequency is, of course, not well defined since no unique
frequency exists. An attempt is made, however, to approximate the frequency as
effected by the regressor value, x, and to force the number of sample points per
"period" to be bounded below by an input parameter. A parameter declaring the
maximum number of samples is also provided which prevents the algorithm from
taking the step size guidance to extremes.
Since all of the integrals are represented as improper integrals, they must be
truncated. An input parameter, e, specifies the greatest absolute truncation error
to be suffered in either integral being computed. Analytically determined cutoff
to
c
CD
<L>
0.5
-0.5

5.5
NUMERICAL TECHNIQUES
269
points, M, are then determined for each integral. By bounding the sinusoidal
terms by unity, M is determined as follows. Solving
/
,-ct _ * CM
ke~ct = -e~cM (5.5.13)
m c
for M results in
M = -iln(f) (5.5.14)
for positive c and k. This implies that the truncation error is no greater than e
when the integral is truncated at M, where
• M = — ln(ae) for r(x)
• M = - i ln(<7,e) for U{x) and W(x).
The cutoffs for s(x) and V(x) require the following lemma.
Lemma5.5.1 Forp > Oandx > max{0,4p(ln(4p) — 1)}, e~xxp < e~x/2.
PROOF: The result follows if it is shown that for such x, \nx < £-. Since
— 2p
In x < x — 1 for all x > 0, we have ln(cx) < ex — 1 for all x, c > 0, in turn
implying that In x < ex - 1 — In c for such x and c. Now, for 0 < c < l/(2p),
we have ex — 1 — lnc < x/(2p) if and only if x > (—1 — lnc)/(l/(2p) — c).
Choosing c = l/(4p) gives the result. I
We consider firstly the cutoff for s(x). Make the change of variables u — ta
and apply the lemma to obtain
/ e-tadt = - / e-V1/")-1^
1 JM"
< -
IM " JM"
J^ae-Udu, q>1,M>1,
al/M-eW2^. <*<1,M*>B,
where B = 4(£ - l) ln[4(i - 1) - l]. Consequently,
a > 1, M > 1,
./A
oo
M I le-M°/2, a<l,M>B^a.
This shows that truncating the integral defining s(x) at M, where
{-ln(a£)}'/Q VI, a>\,
M= [
{-21n(a:e/2)}1/QVSI/Q, a<l

270 NON-LINEAR REGRESSION 5.6
results in truncation error not greater than e.
We now turn to the cutoff for V(x). Use the lemma to write for M >
4o-f'(ln4-l)«1.545afI,
'l
/•oo
/•oo /-oo
/ e-ait(\+\nt)dt < / e-^Hdt
Jm Jm
1 /-OO
< _L / e->t/2dt
a\ Jm
1
Thus, defining M = max{4<7, ' (ln4-1), -2cr, ' ln(cr2e/2)}, the truncation error
incurred by truncating the integral defining V{x) at M is no greater than e.
5.6 Exercises
Exercise 5.1 Let G\ and G2 be i.i.d. N(0,1) and let A ~ 51/2(1,1,0)
independent of Gi and G2. Let X = AX'2GX and Y = AX'2G2. Prove that the joint
probability density function f(x,Y) (x' v) °^ (-^> -H satisfies
f(.x,Y) (x, y) ~ Ci (x)y~2~a as y -► oo,
Hint: Use
f(x,Y)(x>y) = const. / exp|--(x2 + y2)u|-/Af-^u
and
/ e-eu-/A(-W~ const, r
3_1~t as0-> oo.
Exercise 5.2 Suppose in Example 5.1.5 that Yj and Y"2 are i.i.d. Sa(a, 1,0) with
a < 1. Their probability density function is supported on the positive real line.
Show that £(|X2|p|Xi = re,) < oo a.s. for all p > 0.

Chapter 6
Complex stable stochastic
integrals and harmonizable
processes
A (real) harmonizable a-stable process is a stochastic process of the form
/oo
eitxM(dx), -oo < t < oo,
-oo
where M is a complex-valued a-stable random measure on Borel subsets of the
real line with a finite control measure m. A reader familiar with the spectral theory
of stationary Gaussian processes will recall that every stationary Gaussian
process, continuous in probability, can be so represented and hence is harmonizable.
This is not true, however, for stationary a-stable processes with index a 6 (0,2).
Nevertheless, the class of stationary harmonizable a-stable processes is of
importance and it deserves a careful study. Because the definition involves the complex
integral J™ eltxM(dx), we need to introduce complex stable random measures
M and complex-valued integrals
iv) = r f(x)M(dx),
J—oo
which are of interest in their own right. For simplicity of notation and to make our
formulas as short as possible, we restrict our discussion to the SaS case, although
generalization to the skewed case should be possible by using similar arguments.
In Section 6.1 we introduce complex-valued SaS random measures and the
related notions of "circular control measure" and "control measure." Integrals
with respect to complex-valued SaS random measures are constructed in Section

272 COMPLEX STABLE STOCHASTIC INTEGRALS 6.1
6.2. In Section 6.3 we treat the important special case where M is a complex
isotropic SaS random measure. In that case, /(/) is a complex isotropic SaS
random variable. Such random variables were first studied in Section 2.6. Their
real and imaginary parts are identically distributed but, in general, not independent.
They are independent if a = 2.
Complex measures M and integrals 1(f) have also a series representation
which is given in Section 6.4. In Section 6.5 we consider harmonizable processes,
not only the real harmonizable process Re J™ eltxM(dx) but also its complex
counterpart /^° ettxM(dx). The main theorem in the section gives necessary
and sufficient conditions for these processes to be stationary. In Section 6.6 we
consider stationary real harmonizable processes. Although these processes share
properties with stationary sub-Gaussian processes, it is shown that the two classes
are "almost" disjoint.
6.1 Complex-valued SaS random measures
Complex-valued SaS random measures are analogous to the real random
measures defined in Section 3.3, but they are, naturally, a little more complicated.
Let (CI, T, P) be the underlying probability space, and let L°(C1) and L°c(Ci)
be, respectively, the sets of all real and complex random variables defined on
(CI, T, P). Every element of L°c (CI) is of the form X + iY, where X, Y 6 L° (CI).
Let (E, £) be a measurable space, and let (52, Bi) be the unit circle in R2 equipped
with the Borel a-algebra. Let fc be a measure on the product space (E x 52, £ x B2)
satisfying the following condition:
Condition 6.1.1 For every A e £ such that k(A x 52) < 00, k(A x •) is a (finite)
symmetric measure on (S2, #2).
We shall consider
£0 = {A e £ : k(A x 52) < 00}. (6.1.1)
Definition 6.1.2 A complex-valued SaS random measure on(E,£) with circular
control measure fc is an independently scattered a-additive complex-valued set
function
M : £0 -» L°c(Ci)
such that, for every A € £0, M^(A) = Re M(A) and M^ (A) = Im M(A) are
jointly SaS with spectral measure k(A x •).
The meaning of "independently scattered" and "o--additive" is the same as in
Section 3.1, but the reader should-note that this time these notions are applied
to complex-valued random variables, i.e., to the vectors formed by their real and

6.1
COMPLEX-VALUED SaS RANDOM MEASURES
273
imaginary parts. Hence, independence of the complex-valued random variables
M (Ai), M(A2), ■■■, M(Ak) means independence of the random vectors
Ml'HA,) \ / MW(A2) \ ( MW(Ak)
M<2>(A,) )' \ M<»(A2) )'""{ M™(Ak)
One can apply Kolmogorov's existence theorem to show that the complex-valued
SaS random measure M in Definition 6.1.2 exists, with finite-dimensional
distributions
d
Et^UY^i^M^iA^+efM^iAj))^
i=i
= exp{-/ J \J2(s^l) + s2ef))lA.(x)\ak(dx,ds)} (6.1.2)
where Ax, A2,..., Ad e So and (o\l), 9™),{6g\ ^2)),..., (0<°, 0<2)) are pairs
of real numbers (Exercise 6.1).
Notation. k(dx, ds) and k(dx x ds) are equivalent notations.
Remarks
1. We will often express the left-hand side of (6.1.2) as
d
Eexp/iRe ^ZjM.,(A,)}, (6.1.3)
where Zj = O^+iOj , and Zj = #j —i#j denotes the complex conjugate
Of Zj.
2. A complex-valued SaS random measure is characterized by a circular
control measure k defined on the product space E x S2, whereas a real-
valued SaS random measure is characterized by a control measure m
defined only on E.
3. To understand the need for k, think of M(dx) as a SaS random variable
measuring the "size" of a small set around point x e E. Then, in the
real-valued case, the distribution of the SaS random variable M(dx) can
be specified by a single number - its scale parameter m(dx). Thus, in the
real-valued case, M is defined by specifying m on E. In the complex-
valued case, the distribution of the (complex-valued) SaS random variable

274 COMPLEX STABLE STOCHASTIC INTEGRALS 6.1
M(dx) is specified by the spectral measure of its real and imaginary part,
that is, by k(dx, ds) where x e E and where the argument s belongs to
the unit circle 52. Hence, in the complex-valued case, M is defined by
specifying fc on E x Sz-
4. The measure fc is called "circular control measure" and not "control
measure" because the latter term is reserved for the measure fc(- x 52) =
Js^k(-,ds)onE.
Definition 6.1.3 The measure on (E, £) defined by m(A) = k(A x S2) is called
the control measure of complex-valued SaS random measure M.
Although, in the complex-valued case, the control measure m does not, in
general, determine uniquely the (finite-dimensional distributions of) random
measure M, it is a useful object because it is simpler than the circular control measure
fc and because there are properties of complex-valued SaS random measures M
that can be conveniently stated in terms of m.
Example 6.1.4 The subset £0 = {A € E : k{A x S2) < 00} off introduced in
(6.1.1) can be written 60 = {A e £ : m(A) < 00}.
Example 6.1.5 If the circular control measure fc is concentrated on the two "lines"
E x {(1,0)} and E x {(— 1,0)}, then the complex-valued SaS random measure
M is, actually, real, and its control measure m (given, in this case, by m(A) =
2k(A x {1,0}) does determine uniquely the random measure M.
Example 6.1.6 Suppose that the circular control measure fc is a product measure
of the form fc = mj, where m is the control measure of the random measure M
and 7 is the uniform probability measure on the unit circle S2. Then the random
measure M has the property
e^M = M, for any real <j> (6.1.4)
(see Section 2.6), where, as usual, = denotes equality of the finite-dimensional
distributions. We call such a measure isotropic (or rotationally invariant).
Conversely, any complex-valued SaS random measure satisfying (6.1.4) must have
a circular control measure of the form fc = mj (see Exercise 6.2.) Note that
7 must be a probability measure because the relations k(A x 52) = m{A) and
k(A x Si) = m(i4h(S2) imply 7OS2) = 1-
Of course, the finite-dimensional distributions of an isotropic SaS random
measure M are determined uniquely by its control measure m.

6.2 INTEGRALS WITH RESPECT TO COMPLEX SaS MEASURES 275
It is obviously possible to view a complex-valued SaS random measure M
as a pair of two dependent real SaS random measures M^ and M'2', whose
control measures m^ and mP^ are given by
mw{A)- [ \si\ak{Axds), i = 1,2. (6.1.5)
(See Exercise 6.3.) This perspective, however, is often not useful because the
dependence between the random measures M^ and M^ can be very complicated.
Example 6.1.7 A complex-valued SaS random measure M has independent real
and imaginary parts if and only if the circular control measure k is concentrated
on the four "lines" Ex {0,1}, Ex {0,-1}, E x {1,0} andE x {-1,0}. (See
Exercise 6.4.) Unlike the Gaussian case a = 2, an isotropic measure M cannot,
therefore, have independent real and imaginary parts, unless M is identically zero.
(See Exercise 6.5.)
6.2 Integrals with respect to complex-valued SaS
random measures
Having defined a complex-valued SaS random measure M, we shall now define
the integral
1(f) = / f(x)M(dx) (6.2.1)
Jb
of a non-random complex-valued function / with respect to M.
Let / = /O + ifW : E —* C be a measurable function. Since we already
know how to integrate real-valued functions against real-valued SqS random
measures, it is tempting to write / fdM = J(/(1) + i/(2))d(M(1) + iM(2)) and
hence to define the integral in (6.2.1) by
f f(x)M(dx) =([ fV\x)MW(dx) - I }{1\x)M{-1\dx)\
+ i(f fW(x)M®{dx) + I /(2)(a:)M(1)(d a;) I . (6.2.2)
This relation should, of course, hold for any reasonable definition of the integral,
but it is inconvenient to define the integral (6.2.1) by (6.2.2) because the nature
of the dependence between the various integrals on the right-hand side of (6.2.2)
is not readily apparent. A more convenient approach, it turns out, is to use one
of the possible constructions, developed in Chapter 3, for the real-valued case.
We choose the constructive definition discussed in Section 3.4 because it best

276 COMPLEX STABLE STOCHASTIC INTEGRALS 6.2
demonstrates the Wiener-type nature of the integral. Throughout this section,
unless explicitly stated otherwise, all SaS random measures M are complex-
valued.
Let M be a SaS random measure on (E, £) with a circular control measure
k. For a simple function of the form f(x) = ]C™=1 CjIa, (z) where the Cj are
complex numbers and the AjS are disjoint sets in fo, we define
1(f) = [ f(x)M(dx) = £c,-M(A,-). (6.2.3)
It is trivial to check that for simple functions, 1(f) is linear in / and satisfies
(6.2.2) a.s. The real and imaginary parts of 1(f), denoted henceforth I^(f) and
1^ (/) respectively, are jointly SaS with joint characteristic function
Eexj>{WJW(f) + 02ll2)(f))}
= £?exp{t(*i[ f fw(x)M^(dx) - f f{2)(x)MW{dx)]
' +fc[/ fm(x)M^(dx)+J fM(x)M™(dx)])}
+ I Uhf«\x) - exfV(x))MW(dx)\ }
= exp{-/ f\el(sxf^(x)-s2fV\x)) -
1 JB J Si'
+02(sif{2)(x) + s2fW(x))\ak(dx,ds)}, (6.2.4)
by Relation (6.1.2). Note also that
SexpWfl.jOUJ + fclPH/))}
= exp{- J J |(a,0i + s202)fW(x) + (Sle2 - S2el)ff-2Hx)\ak(dx,ds)}.
(6.2.5)
Having defined the integral for simple functions, "we are ready to apply the
"approximate-and-use-convergence-in-probability" argument. We will see below
(Proposition 6.2.2) that, as in the real-valued case, the class of functions / for
which the integral /(/) can be defined is
La(m) = {/:£-* C : f \f(x)\am(dx) < coj,
J E

6.2 INTEGRALS WITH RESPECT TO COMPLEX SaS MEASURES 277
where m is the control measure of M and |/| denotes the modulus of /.
Formally, for any / = f^ + ifW S La(m), choose a sequence of simple
functions {fn.}^Li such that
/n(x) —> f(x) for almost every x € E, (6.2.6)
\fn(x)\ < 9(x) for any n, x and some 9 £ La(m). (6.2.7)
Such a sequence always exists as we can take, for example, fn{x) = fn (x) +
if^(x) with
' i if i</O)(*)<i£V* = 01lI...ln2-lI
/„0)(i)=| -£ if-^</O)(a:)<-i,i=0,l>...>n2-l>
. 0 if |/W)(i)|>n,
for j = 1,2,..., in which case 9 = |/|. The sequence of integrals I{fn), n =
1,2,..., is well defined by (6.2.3). Note that by (6.2.4), for any n, m, J^ (/„) -
i"(1)(/m)~ SQ-(<#L0,0), where
(41!jq = / / ^.(/(''(^-/^(^-^(/^(^-/^(^rfc^ds).
Relations (6.2.6) and (6.2.7) imply an,m —* 0 as n,m —> oo and hence the
sequence {1^ (fn)}^Li converges in probability. The sequence {^(/n)}^=i
converges in probability as well. Therefore, the sequence of integrals {I(fn)}%Li
converges in probability and we define
1(f) = plim^/t/n),
and write /(/) = fE fdM. Arguing as in Section 3.4, the integral 1(f) does not
depend on a particular choice of approximating sequence {/„ }^L,. We summarize
its properties in
Proposition 6.2.1 (i) For any f e La(m), the integral 1(f) is a complex-
valued SaS random variable satisfying (6.2.2), with characteristic function given
by Relation (6.2.4).
(ii) For any /, <? € La(m), a, b complex constants,
I(af + bg) = al(f)+bl(g) a.s.
(Hi) For any f\, f->,-.-, fd 6 LQ(m) and for any pairs of real numbers
(»{,,.fl?)),(^),fli2)),...,(^),fl?)).
- biEexP{iJ2(0yiWVi) + 0f/(2)(/i))} (6-2-8)

278
COMPLEX STABLE STOCHASTIC INTEGRALS
6.2
= / / \i2{0f)(slfji\x)-s2f^(x))+ef)(3lfj2\x)+a2fjl)(x))}Mdx,dS)
Je Js2'j-l
= [ [ ElM?5 + s2ef])/j!)(x) + (s.flf -S2el1))fj2)(x)} ak(dx,ds).
JEJs2'j=t
PROOF: (i) /(/) is obviously a SaS random variable, being a limit in probability
of a sequence of SaS random variables. The characteristic function of 1(f) is
given by (6.2.4) because the characteristic functions of the I(/n) have this form,
and convergence in probability implies convergence in distribution and, thus,
pointwise convergence of characteristic functions. Finally, (6.2.2) holds for the
/(/„), and every one of the five integrals involved converges in probability to the
corresponding integral of/. Therefore (6.2.2) holds for /(/).
(ii) The argument is identical to the one in the real-valued case and is left to
the reader.
(iii) For j — 1,2,..., d define
Uj = —
4° + ef
e^-ef
aj= uj + *uj-
By part (ii) of this proposition,
j=\ j=l
j=i
i=i
Therefore,
Ec^i^o^i^if^ + efh^ifj))}
= £exP{z(/0)^a./.)+/(2)(^aj/.))}
. . d d
= exp{- / / I [s, Re (jTajf^x)) ~ s* Im (S0^*))]
+ si Im QT 0,^(2;)) + s2 Re (]T]aj/j(x))] | Hdx,ds)j,
j=i

6.2 INTEGRALS WITH RESPECT TO COMPLEX SaS MEASURES 279
by (6.2.4). Rearranging terms yields {6.2.8). I
In the real-valued case, the form of the characteristic function of the stable
integral (see Proposition 3.4.1) makes it apparent that the integral can only be
defined for functions in La(m). In the complex-valued case, the form of
characteristic function (see (6.2.4)) is more complicated, but it is still true that the
integral cannot be extended outside of La(m). Indeed,
Proposition 6.2.2 Let f : E —* C be a measurable function such that there is a
sequence {/n}^Li of measurable simple functions satisfying (6.2.6) and such that
the sequence of integrals {I(fn)}%L\ defined by (6.2.3) converges in distribution.
Then f £ La(m).
PROOF: By (6.2.4), I<0(/n) ~ SQ{an,0,0), where o% = JE /ft h/^(x) -
s2fk2){x)\ak(dx,ds). Since {/<'>(/„)}£,, converges in distribution, conclude
that limn-^oo a% exists and is finite. By Fatou's lemma,
/ f \sjM(x)-s2fW(x)\ak(dx,ds)
Jb Js2
< lira /" / \sJ^{x)-s2f^(x)\ak(dx,ds)<oo. (6.2.9)
n—°° Je Js2
Similarly, {/^2^(/n)}^L, converges in distribution and, hence,
\sj(2)(x) + s2fw(x)\ak{dx,ds) < oo. (6.2.10)
//
JE J Si
IEJS2
Now by (6.2.9) and (6.2.10), and Fubini's theorem,
f \f(x)\am(dx) = [ /"(/<•>{x)2 + fV)(x)2)a'2k{dx,ds)
JE JE JS7
= / f [(Slf^{x)-s2f^(x))2 + (sJ^(x) + S2f^(x)2)}^2k(dx,ds)
JEJ Si
< [ f [\slfM(x)-s2fW(x)\« + \sjW(x) + s2fW(x)\a)k(dx,ds),
Je Js2
which is bounded. 1
Many properties of stable integrals in the real-valued case have counterparts in
the complex-valued case. The following result, for example, should be compared
to Proposition 3.5.1 which gives the corresponding result in the real-valued case.
Proposition 6.2.3 Let Xj = JEfj(x)M(dx), j = 1,2,..., and X =
Je f{x)M(dx), where M is a SaS random measure with circular control
measure k and control measure m, and /, fj, j = 1,2,..., are in La(m). Then
Plimj-oo*j = X>

280 COMPLEX STABLE STOCHASTIC INTEGRALS 6.2
if and only if
lira / \fj(x) - f{x)\am{dx) = 0. (6.2.11)
Proof: Note that plim^^Xj = X is equivalent to plimJ_<0O(Xj - X) = 0,
which is equivalent to plim^^ Re (Xj -X) = 0 and plim^^ Im (Xj -X) = 0.
This, in turn, is equivalent to Re (Xj - X) 4- 0 and Im (X, - X) 4- 0, where
4 denotes convergence in distribution. By (6.2.4), Re(X, - X) ~ S^ct^, 0,0)
and Im(X,- - X) ~ 5Q (erf}, 0,0) where
(^'))a= / / l*i(/j')(a:)-/(,)(a:))-«2(/!2)(a:)-/(2)(x))rA:(dx>ds))
(^2>)a= / / ls>(/f)(a;)-/(2)(x))+S2(/j,)(x)-/(1)(a;))|^(dX)ds).
Therefore plim •_00Xj = X is equivalent to
lim v™a = lim cf)q = 0. (6.2.12)
j—>oo 3
Trivially, (6.2.11) implies (6.2.12).
On the other hand,
JE
II
II
JE JS-,
\fj(x) - f(x)\am(dx)
JE
" [(/fCz) - /(1)(*))2 + (/f(x) - fV(x))2r'2k(dx,ds)
2
[(Sl(f(jl)(x) - /<•>(*)) - s2U?\x) - fW(x))f
2
+(*i(/j2)(z) - /(2)(*)) + s2(fjl)(x) - f^(x)))2)^2k(dx,ds)
so that (6.2.12) implies (6.2.11) as well. I
Let M be a SaS random measure on (E, £) with circular control measure k,
and let / e La(m). Since this book deals mainly with real-valued stable laws,
we will encounter I^\f) = Re /(/) more frequently than /(/). Of course, the
properties of I^(f) can be deduced from those of 1(f). For future reference, we
collect some useful facts about I^(f).
Proposition 6.2.4 (i) J(1)(/) ~ SQ(crf, 0,0), where
a?= f [ \sif(,)(x) - s2f<-2\x)\ak(dx,ds).
JE JS2

6.3
THE COMPLEX ISOTROPIC SaS CASE
281
(ii) For any /, g € LQ(m) and any real constants a, b,
I^(af + bg) = a/<l)(/) + bl^{g) a.s.
Example 6.2.5 The {real) harmonizable SaS process is defined as
/oo
eitxM(dx), -oo < t < oo, (6.2.13)
-oo
where M is a SaS random measure on (R, B) with a finite circular control measure
k (equivalently, with a finite control measure m). Proposition 6.2.4 and Theorem
2.1.5 show that it is a SaS process. We shall investigate some of its properties in
Sections 6.5 and 6.6.
6.3 The complex isotropic SaS case
We treat in this section the important special case where the complex random
measure M is not only SaS but also isotropic, that is, its circular control measure
is of the form
k = m7,
where 7 is the uniform probability measure on the unit circle S2 and where m, the
control measure of M, is a measure on (E, S).
Let / be complex-valued with
/ \f{x)\am{dx) < 00
Je
and consider the complex-valued integral
1(f) = [ f(x)M(dx),
Je
where M is a complex isotropic SaS random measure with control measure
m. Let I^{f) and I^(f) denote, respectively, the real and imaginary parts
of !(/). These are real-valued random variables. The following theorem gives
the characteristic function of the complex-valued random variable /(/), i.e., the
characteristic function of the vector (i^'H/)) J(2)(/)).
Theorem 6.3.1 Let M be complex isotropic SaS, 0 < a < 2 with control
measure m. Then for any complex number z = 9\ + iQj,
Ecxp{i(8lI^){f) + e2I{2)(f))} = £exp{iRez/(/)} = £exp{i Re/(:?/)}

282 COMPLEX STABLE STOCHASTIC INTEGRALS 6.3
where
cxV{-\z\acoJ \f(x)\am(dx)}, (6.3.1)
i r
co = — / |cos<j!>|Qd<?!>.
Proof: Using Relation (6.2.4) and k(dx, ds) = m(dx)f(ds), we have
- In £exp{i(0, I^(f) + 02/(2)(/))}
= / / \el(slf^(x)-s2f^(x)) + e2(slf^(x)+S2f{l)^Wm(dxHds)
Jbjsi
= 11 \(suS2)ielf^(x)+62f^(x),-6lf^(x)+e2f^(x))rm(dxMds).
JeJsi
Setting5 = 5(1)+i3(2), 9{i) = Oif{1){x) + 02/(2)(*) andg<2> - -0,/<2>(aO +
^2/^Hx)> we obtain
£?CXp{i(fl,/^(/) + 02/«(/))}
= exp{- f [f \si9W(x) + s29W(x)\a1(ds)\m(dx)}
= exp{-co7(52) J \g(x)\am(dx)}
as in the proof of Corollary 2.6.5. But j{S2) = 1 and \g\2 = {g^)2 + (g&)2
equals
02(/«)2 + 8l(f{2))2 + 2^^2/(1)/(2) + 02(/(2))2 + W>)2 - 2^,^2/(1)/(2)!
i.e., |<?|2 = (0? + 6\)\S\2. Therefore
£exp{i(0,I(,)(/) + ^2/(2)(/))} = exp{-(02 + 92r'2cc J \f(x)\am(dx)}
= exp{-\z\acoj \f(x)\am(dx)). I
Remark. We normalized 7 by 7(^2) = 1. Had we normalized it by 7(52) = c^\
the constant Co would not appear in (6.3.1).
Isotropic SaS random variables were introduced in Section 2.6. Comparing
the statements of Theorem 6.3.1 and Corollary 2.6.5, we note that if M is isotropic
SaS, then the random variable /(/) is isotropic SaS as well.

6.3
THE COMPLEX ISOTROPIC SaS CASE
283
One advantage of dealing with isotropic SaS random measures or integrals is
that we can ignore the circular control measure k and use exclusively the control
measure m. Moreover, form (6.3.1) of the characteristic function is very similar to
that of characteristic functions of real-valued SaS integrals. The only difference
is the presence of the constant cq and that of \z\a — (9\ + 6\)al2 which appears
in the exponent instead of \9\a. By analogy to the real case, we will sometimes
express the characteristic function of 1(f) = JE f(x)M(dx) as
EtxV{i(9J^(f) + 02/(2)(/))} = EeiRe '^ = e-"Re *F/)II2,
where z = 9\ + i92. Relation (6.3.1) then becomes
||Re I(zf)\\% = \z\°cq f \f(x)\am(dx) = \z\°*co\\m, (6-3.2)
Je
Theorem 6.3.1 provides also information on the real and imaginary parts of
1(f).
Corollary 6.3.2 For any real 9,
£exp{i5»7(1)(/)} = £exp{i0Re / f(x)M(dx)\
= exP{-co\9\a J \f(x)\am(dx)} (6.3.3)
and, moreover,
I{2)(f) = IW(f).
Proof: In Theorem 6.3.1, set first 6{ — 6 and 62 = 0 and then set 6\ = 0 and
e2 = e. ■
The real and imaginary parts of /(/) are identically distributed but they are,
in general, not independent when 0 < a < 2. This follows from Section 2.6
because /(/) is a complex isotropic SaS random variable. But it can also be
directly observed here. Indeed, for z = 9\ + id2 7= 0 and ||/||Q 7^ 0,
Eexp{i(0,/<">(/) +02/(2)(/))} ± {Eexpi0ilW{f))(Eexpid2lV>{f)),
because
(9] + 0f)Q/2co||/||S ± |0,rco||/||° + \92\aco\\f\\aa.
There is equality, however, in the Gaussian case a = 2, and in that case, the real
and imaginary parts of /(/) are independent.
The next corollary shows that a linear combination of the real and imaginary
parts of /(/) has, up to a multiplicative constant, the same distribution as the real
partof/(/).

284
COMPLEX STABLE STOCHASTIC INTEGRALS
6.3
Corollary 6.3.3 Let A and B be real. Then
AIM(f) + BlW(f) I (A2 + B2)l'2lW(f).
Proof: For any real 9,
Ecxp{iB(AlW{f) + BlW{f))} = tx?{-c<s{91A2 + e2B2)a>2\\f\\Z}
= expf-comK^ + B2)1'2/^}
by Theorem 6.3.1. Use now Relation (6.3.3). I
The following theorem is an extension of Theorem 6.3.1 to stochastic
processes.
Theorem 6.3.4 Let {I(ft) = JEft(x)M(dx), t € T} be a complex-valued
stochastic process where M is an isotropic SaS random measure with control
measure m, and let 1^ and 1^ denote, respectively, the real and imaginary parts
of I. Fix t\,...,td € T. Then for arbitrary complex numbers Zj = dy + iQ\ ',
j = l,.-.,d,
£exp{i(X:^)/(1)(/ti)+^2)/(2)(/ti))} = £exp{i]T Re/&•/*,)}
i=i j=i
r d
= exp{-co / | J^ZjfaWmidx)}. (6.3.4)
'* i=i
Proof: Apply Theorem 6.3.1 with 0, = 1, 62 = 0 and / = Y?j=i Zjfti. ■
Remark. Theorem 6.3.4 is often expressed as follows:
d d
£exp{iRe £7(2,-/*,)} =exp{-||Re £'&/*,)IIS}.
j=i j=i
where
Re 21(zjftj) = co / | £ V*i(x)|Qm(dx) = Co E^/i,^
(6.3.5)

6.3
THE COMPLEX ISOTROPIC SaS CASE
285
Corollary 6.3.5 Let {I(ft), t £ T} be defined as in Theorem 6.3.4. The joint
characteristic function of Re I(ft), t e T is
d . d
£exp{i ]T 9j Re J(/tj)} = exp{-co / | £ 9jftj(x)\am(dx)\ (6.3.6)
and
{Im7(/t), *6T}^{ReJ(/t), t e T}.
Remark. The major difference between the joint characteristic functions of the
complex-valued process {I(ft), t € T} and those of the real-valued process
{Re I(ft), t € T} is the presence of the complex numbers Zj instead of 9j in the
characteristic functions.
Example 6.3.6 Let M be an isotropic SaS random measure on R with control
measure m and consider the complex-valued process
Assume
/oo
eitxM(dx), t€R.
-OO
/CO
\eitx\am(dx)=m(R) < oo
-oo
in order to ensure that the process H is well denned. Its joint characteristic
function is
EexpjiX^tfWte) + 0ftf(2>fe))} (6-3.7)
where H^(t) and H^(t) are, respectively, the real1 and imaginary
prats of H(t). By Theorem 6.3.4, that characteristic function equals
exp{-cd/Z^|i:J=5l2je"'T"»(<kO} where Zj = 0^ + i6?\ j = 1.....A
Now,
d d 2 d d
= [^B^costjX + ^e^siatkX +|5^fl}l)sintji-53fl{.2)costfci
j=i fc=i j=i fc=i
d d
= ^Yl^9^ +ef)e'k)) (costjXCOstkx + smtjxsintkx)
j=i fc=i
'#('>(«) = Re P° eit:rA/(da;) was considered in Example 6.2.5, but we did not suppose that
the random measure M is isotropic, as we do here.

286 COMPLEX STABLE STOCHASTIC INTEGRALS 6.4
d d
+ 2 Y^ Yl i9^8^ cos lix sin hx - Oj^O^ sin tjx cos tkx)
j=\ fc=i
+2EE^1)^2)sin^-^)x-
Therefore the finite-dimensional characteristic function (6.3.7) of {H(t), igR}
equals
-oo d d
exp{-co / £BW} + WW* -*i)»
d d i.
+ 2^^^.1)42)sin(tfc-ti)x|Q m(da;)} (6.3.8)
i=i fc=i
The process .ff is the complex-valued harmonizable process and we will encounter
it again in the sequel.
Setting 6j = &j and 0j ' — 0, we obtain the finite-dimensional characteristic
function of {Re H{t), t € R}:
d
Eexp/i^Tflj Re #(*.,)}
|££0.Acos(t,--tfc)z m(cte)}. (6.3.9)
j=l fe=l
6.4 Series representation of complex-valued SaS
random measures and integrals
The series representation of real SaS random measures and stable stochastic
integrals developed in Chapter 3 is extended here to the complex-valued SaS
case.
Let M be a complex-valued SaS random measure on (E, S) v/ithfinite circular
control measure k and control measure m. Then m(E) = k(E x 52) is finite
and is both the total mass of k and the total mass of m. Thus k = k/m(E) is a
probability measure on {E x S2, S x B2). Let {T], r2,...} be a sequence of arrival

6.4
SERIES REPRESENTATION OF COMPLEX INTEGRALS
287
times of a Poisson process with unit arrival rate and let {ei, e2, • ■ •} be a sequence
of Rademacher random variables. Finally, let {Vi, V2,...} be a sequence of i.i.d.
(E x S2)-valued random variables with common law k. All three sequences,
{H, r2, • • ■}, {et, e2,.. •} and {Vj, Vz,. ■.}, are assumed to be independent. We
write Vj = (^-.(S^Sf)), 3 = 1,2,..., with Z, € E, (SJ'^sf) € S2.
We have the following:
Theorem 6.4.1
{M(A), A G £}
CO
i {(CQm(E))1/Q^eir-1/Ql(Zj e A)(^j!> + iS<2)), A 6 f}, (6.4.1)
i=i
w/zere CQ is given in (1.2.9).
We will not prove this theorem, as it can be regarded as a particular case
of Corollary 6.4.3 below, involving series representation for stochastic integrals
with respect to complex-valued SaS random measures. We stated Theorem
6.4.1 because we feel it is the structure of a random measure which determines
the structure of an integral with respect to that measure. But, in fact, a statement
about a random measure follows from the corresponding statement about integrals
with respect to that measure.
Theorem 6.4.2 Let f : E -* C belong to La(m). Then
/ f(x)M(dx) L (Cam(E))^J2^rJl/a^Z^l) +i5i2))> (6A2)
Je j=i
where Ca is given in (1.2.9). The series in the right-hand side of '(6.4.2) converges
a.s.
Proof: The a.s. convergence of the series on the right-hand side of (6.4.2) is
equivalent to a.s. convergence of its real and imaginary parts. We shall verify only
the convergence of the real part of the series as the argument for the imaginary
part is identical. Let Uj = Re(/(Zi)(SJ1) + iSf)))1 3 = 1,2,... . Then
1^1 < l/^XSJ0 + »S}2))| = \f(Zj)\, so that £1^1° < 00. Applying
Theorem 1.4.2, we conclude that the real part of the series on the right-hand side
of (6.4.2) converges a.s.
Let 8^ and #(2' be arbitrary real numbers. From Proposition 6.2.1 we
conclude
0M Re / f(x)M(dx)+e{-2) Im / f(x)M{dx) ~ Sa(cr,0,0),
Je Je

288 COMPLEX STABLE STOCHASTIC INTEGRALS 6.4
where aa equals
/ / \eW(SlfW(x) - s2fW(x)) + 6W(SlfW(x) + s2f{l)(xWHdx,ds).
Je Js2
(6.4.3)
To prove (6.4.2) we need to show that the series
CO
(C(a)m(E))X/a Y,eir7'/a [*0) Mf(Zi)(S^ + iS™))
+ 0« 1m{f(Zj)(S^ + tSf))] (6.4.4)
has the same distribution. Note that
OW Re (/(Zj)(5J,) + iSf])) + OM Im (/(^(SJ0 + iSf>))
= ^1)(5<1)/(1)(^) - SJ2)/(2)(2») + ^(SJ^/WCZ,-) + S<2)/(1)(^)),
and thus, applying Theorem 1.4.2, we conclude that the series in (6.4.4) has also
a Sa(a, 0,0) distribution with a given by (6.4.3). This proves (6.4.2). I
Because both the integral on the left-hand side of (6.4.2) and the series on
its right-hand side are linear in /, we can, as in the real-valued case, extend
immediately the result to SaS random vectors and to SaS random processes.
Corollary 6.4.3 Let {ft, t € T} be a family of functions from E into C such that
ft € La(m)for every t € T. Then
{J ft(x)M(dx), t e r}
oo
L {(Cam(E))i^Y,e^Jl/aMZj)(Sf)+iSf)), t € r}.(6.4.5)
j=\
PROOF: Use Theorem 6.4.2 and the linearity to show that every linear
combination of finitely many of random variables on the left-hand side of (6.4.5) is equal
in distribution to the corresponding linear combination of random variables on the
right-hand side of (6.4.5). I
We now turn to a series representation for stochastic SaS integrals with respect
to an isotropic complex-valued SaS random measure. The importance of this
particular case will become evident in the next section in the context of SaS
harmonizable processes.
Let M be an isotropic SaS random measure on (E, £) with a finite control
measure m. We remind the reader that the circular control measure of M is then
given by k = mr/, where 7 is the uniform probability measure on S2. Let f : E —*

6.4
SERIES REPRESENTATION OF COMPLEX INTEGRALS
289
C belong to La(m). Theorem 6.4.2 of course applies. Note, however, that in this
particular case the sequence of S2-valued random variables {(5J ,5- ), j =
1,2,...} is a sequence of i.i.d. random variables with common distribution 7 and
is independent of the sequence of ^-valued i.i.d. random variables {Zj, j =
1,2,...} with common law m/m(E). Let {G^\ j = 1,2,...}, i = 1,2, be two
independent sequences ofi.i.d. standard normal random variables. Then for each
j=\,2,...,
( r'1' r& \
{((G^ + iGf^yn'iiG^y + iGfY)^) (j'j)
(see Exercise 6.11). If we also assume that these normal sequences are independent
of the sequences {Ti, r2,...} and {Z\,Z2,...}, then, by Theorem 6.4.2,
/.
°° G(1)+iG(2)
f(x)M(dx) L {CammXlaYjl"aMti " L-
1*' K ' K K " U ° 3\{Gf? + {Gfyyn
(6.4.6)
Note that the Rademacher sequence {ei,e2,...} has been absorbed in the
(symmetric) normal sequences {G\ ,G[ ,•••}, i = 1,2, since eG = G
if e and G are independent. How can one now eliminate the
denominators Wj = ((G$°)2 + (G^2)))'/2? The sequence {Wj, j = 1,2,...} is
independent of the sequences {Tj, j = 1,2,.. -},{Zj, j = 1,2,...} and
{((G^)2 + (Gffr^HG^ ,Gf), j = 1,2,...} (Exercise 6.11).
Moreover, if ba = EWy, then Wj/ba has ath moment equal to 1 and, therefore, by
Theorem 1.4.2,
/.
00 yj G- + iG^
f{x)M{dx) = (Cam{E)y'aYr-1/af{Zj)^- jrr-1 3m -
v ^ J blJa ((G^y + (Gf]yy/2
00
£ (CabZlm{E))l'a X)r7,/a/(^)(Gi° + *Gf )•
i=i
(Wj has a Rayleigh distribution. Its density function is xe~x /2l(x > 0) and
therefore bQ = EWf is finite. It is easy to verify that ba = 2a/2T{l + §).) This
establishes
Proposition 6.4.4 Let M be an isotropic SaS random measure on (E, £) with a
finite control measure m. Let {G^\ j = 1,2,...}, i = 1,2, be two sequences
ofi.i.d. standard normal random variables, {Tj, j = 1,2,...} be a sequence of
arrival times of a unit rate Poisson process and {Z\,Z2 ...} be i.i.d. E-valued

290 COMPLEX STABLE STOCHASTIC INTEGRALS 6.4
random variables with the common law m/m(E). These four sequences are
assumed independent. Let f : E —> C belong to La(m). Then
(») / f(x)M(dx) (6.4.7)
Je
oo
± (cab-im(E))i/aj2rJ1/anz^GJl)+iGf))'
(n) Re / f(x)M(dx) (6.4.8)
oo
Wien? 6a = 2a/2r(\ + %)andCa is given in (1.2.9).
Remarks
1. Since the series representations of the integrals in Proposition 6.4.4 are linear
in the integrand /, they extend immediately to SctS processes represented as
integrals with respect to an isotropic SaS random measure. See Corollary
6.4.3.
2. It seems at first glance that one could easily simplify the series representation
(6.4.8) by observing that for any j = 1,2,...,
PKz^p + ^{z^cf ± {^\Zjf + fW(Zj?y/*Gf,
and thus
Re / f{x)M{dx)
Je
oo
£(cab-^m(E)y/"J2rJ1/aG<jl)(fw^2 + f{2)(z^y/2-
Indisputably, this series representation requires only one sequence of i.i.d.
standard normal random variables as compared to the two sequences
required in the representation (6.4.8). But, unfortunately, it is not linear in the
integrand / and therefore may not be directly extended to SaS processes
that are represented as integrals with respect to an isotropic SaS random
measure. Thus, in general,
JRe j ft(x)M(dx), t € t}

6.5
HARMONIZABLE PROCESS
291
d °°
* {(CQ&-Im(£)),/oEr7'/°GS,)(/«(1)(^)2 + /«2)(^)2)I/2. *€T}.
This is why we use the representation (6.4.8) instead.
6.5 Harmonizable process
The (real) SaS harmonizable process (6.2.13) has been defined in Example 6.2.5.
Because of its importance in the theory of stationary SaS processes, we shall study
its properties in more detail. It is convenient to work simultaneously with the real
harmonizable process X denned in (6.2.13) and with its complex counterpart
/oo
eitxM{dx), -oo < t < oo. (6.5.1)
-oo
We want to obtain conditions for stationarity. A real stochastic process
{X(t), t e 1} is called (strictly) stationary if its finite-dimensional
distributions are invariant under shift, i.e., for any t\, t2, • ■ •, *d € R and any h > 0,
(X(h),X(t2),..., X(td)) = (X(U + h),X(t2 + h),.. .,X{td + h)).
Similarly, a complex-valued stochastic process {Z(t), t 6 R} is (strictly)
stationary if all finite-dimensional distributions are invariant under shift, i.e., for any
ti,t2,...,td€ Rand any h > 0
(ReZ{tj),lmZ(tj),j = l,...,d) = (ReZ(ti+/i),ImZ(tj+/i), j = l,...,d).
Obviously, if two harmonizable SaS processes, one real-valued, the other
complex-valued, are defined through the same random measure M, then the
real-valued process is stationary if the complex-valued process is stationary, but
the converse is not necessarily true.
The goal of this section is to obtain conditions for the stationarity of the
real- and complex-valued harmonizable SaS processes. Let us start with some
heuristics and try to guess these conditions. Clearly, they must be expressed in
terms of the circular control measure k of the random measure M. We begin with
a simple observation: the behavior of the circular control measure k on {0} x S2
is of no importance as far as stationarity of either the real- or complex-valued
harmonizable SaS processes is concerned. Indeed, let M\ and M2 be two SaS
random measures on R, with circular control measures k\ = (1 — l{o}xS2)k anc*
k2 = l{o}xS2fc> respectively. M\ and M2 are independent because their supports
are disjoint. Let
/oo
e^Mjidx), -oo < t < oo, j = 1,2.
-OO

292 COMPLEX STABLE STOCHASTIC INTEGRALS 6.5
Then Z\ and Z2 are independent complex-valued SaS harmonizable processes
such that {Z\(t) + Z2(t), -co < t < 00} = {Z(t), -co < t < 00}. But Z2
is a constant SaS process (see Exercise 6.9). Therefore, Z is stationary if and
only if Z\ is stationary, and Re Z is stationary if and only if Re Z\ is stationary.
Thus, we should not be concerned with the form of the circular control measure
k on {0} x S2, and we may, and, indeed, shall assume throughout the rest of this
section that the control measure m of the SaS random measure M in (6.2.13) or
in (6.5.1) has no atom at the origin.
In the Gaussian case a = 2 the (complex-valued) harmonizable process (6.5.1)
is stationary if and only if the two components M'1' and M^ of the complex-
valued Gaussian random measure M are independent and identically distributed.
We may be tempted to guess that the same is true in the case 0 < a < 2, but this
turns out to be false. This does not mean that we should not try to extrapolate from
the Gaussian case; it only means that we considered the wrong property. In the
Gaussian case, a complex-valued random measure has independent and identically
distributed components if and only if it is isotropic in the sense of (6.1.4). It is
the property of isotropy that should be carried over to the case 0 < a < 2. We
know from Example 6.1.6 that a SaS random measure M with 0 < a < 2 is
isotropic if and only if k = 1717, where 7 is the uniform probability measure on S2.
This explains heuristically the conditions for stationarity of the complex-valued
harmonizable SaS process that appear in the following theorem.
It should come as no surprise that M does not have to be fully isotropic in order
for the real harmonizable SaS process to be stationary. To express the conditions
for stationarity in the real case, we need to introduce some new notation.
Given a measure k on (R x 52, B x B2), define the conjugate measure k by
k(A xU) = k((-A) x U*), (6.5.4-
A & B, U € Bi and U* ~ {(sus2) G S2 : (s\,-s2) € U}. In other words, I
is the image of k under the measurable map h : 1 x S2 ~» t x S2 defined by
h({x,si,s2)) = (—x,s\,~s2). Similarly, for a measure m on (R,<8), we define
the measure m as m(A) = ro(—A), A € B. We are now ready to state the main
theorem.
Theorem 6.5.1 Let M bea SaS random measure on (R, B) with a finite circular
control measure k and a control measure m, and suppose that m has no atom at
the origin. Then the complex-valued harmonizable SaS process Z in (6.5.1) is
stationary if and only if the SaS random measure M is isotropic (rotationally
invariant), i.e., if and only if it satisfies
k — 7717,
(6.5.3)

6.5
HARMONIZABLE PROCESS
293
where 7 is the uniform probability measure on S2.
The real harmonizable SaS process X in (62.13) is stationary if and only if
the SaS random measure M satisfies the weaker condition
k + k = (m + m)7- (6.5.4)
PROOF: Let ks = k + fc and let Ms be a SaS random measure on (M, B) with
ks as circular control measure. Then the control measure ms of Ms satisfies
ms =m + m. Define
/oo
eitxMs{dx), -00 < t < 00. (6.5.5)
-00
It is trivial to check using Proposition 6.2.4 that
{Y(t), -00 < t < 00} = {2l/aX{t), -00 < t < 00}. (6.5.6)
Therefore, X is stationary if and only if Y is stationary.
Proof of the necessity part of Theorem 6.5.1: We use two steps.
Step 1. We assume that the process Y (respectively, the process Z) is stationary,
and we prove that the SaS random measure MT denned by
MT{A) = [ eiTXMs{dx), AeB
Ja
(respectively, defined by MT(A) = JA elTXM(dx), A £ B) has the same
distribution as Ms (respectively, M) for every real r.
Step 2. We assume that MT has the same distribution as M, for every real t,
and prove that the SaS random measure M is isotropic.
These two steps imply the necessity part of the theorem. Indeed, if the process
Z is stationary, then Step 1 and Step 2 imply that the SaS random measure M is
isotropic, and then (6.5.3) follows from Example 6.2.5. Similarly, if X (and thus,
Y) is stationary, then the same argument applies to Ms, and (6.5.4) follows. We
start with the
Proof of Step 1. It is well known from Weierstrass' trigonometric approximation
theorem that for any real continuous periodic function / with period 2-n and for any
e > 0, there is a trigonometric polynomial of the form T(x) = l]"_0(aj cos jx +
bj sin jx) such that | f(x) - T(x) \ <e for every real x. It follows immediately that

294 COMPLEX STABLE STOCHASTIC INTEGRALS 6.5
if / is a continuous periodic function with period R, then there is a trigonometric
polynomial of the form
T(x) = tl{ai C0S(^J'S) + bj S]n(~fijX)) (6"5"7)
such that | f(x) - T(x) | < e for every real x. Obviously, if / is even (respectively,
odd), then we may always choose bj = 0, j = l,...,n (respectively, a,- =
0, j=l,..., n) in (6.5.7).
Letnowpbeacomplex-valuedfunctionoftheformg = <?0-H<^2\ where <?(')
(respectively, g^) is an even (respectively, odd) continuous periodic function with
period R, and let e > 0. It follows from the preceding remarks that there are two
trigonometric polynomials T(1)(x) = £"=0a.jcos((2n/R)jx) and T(2)(x) =
J2%0bjS\n{(2T:/R)jx) such that \T^(x) - gW{x)\ < § for any x S R,
k = 1,2. Choosing e = 2~m, m = 1,2,..., we conclude that there is a
sequence of functions of the form
9m{x) = Yy^ieWW* + e-i(2x/R)Jx) + ih(rn)(eiV*/R)jx _ e-i(2„R)jxy
(6.5.8)
with real-valued coefficients {aj , bj, j = 0,1,... ,nm} such that for every
m = 1,2,... and for every real x we have \g(x) — gm(x)\ < 2~m. For any
function g and its approximating sequence {gTO}ro=i as above we define
/CO /-CO
g{x)Ms(dx), Ym = Re / gm(x)Ms(dx), m = 1,2,....
-co J—oo
Assume, now, that the real harmonizable SaS process Y(-) defined in (6.5.5)
is stationary, and let r be a real number. For MT given by MT(A) —
JA eiTXMs{dx), AeB, define
/CO /-OO
g{x)MT(dx), yT,m = Re / gm{x)MT{dx), m = 1,2,....
-CO J — CO
By the stationarity of Y (•), we have
{Y{t), -oo < t < oo} = {Y{t + r), -oo < t < oo}
= JRe / ei(t+T)*Ms(dx), -oo < * < oo}
^ J—CO
/•CO ^
= |Re / eitxMT{dx), -oo < t < oo},

6.5
HARMONIZABLE PROCESS
295
where Exercise 6.7 has been used in the last equality. Hence, by Proposition 6.2.4,
Ym = Yr,m- Proposition 6.2.3 implies plim^^F™ = Y, plimm_00YTtm = YT.
Therefore,
r°° w r°°
Re / g(x)Ms(dx) = Re / g(x)MT(dx) (6.5.9)
J ~oo J—co
for any function g:l->C whose real and imaginary parts are, respectively, even
and odd continuous periodic functions.
Our next goal is to extend the class of functions g for which (6.5.9) holds.
Let g be a continuous function with bounded support whose real and imaginary
parts are, respectively, even and odd. Choose K large enough so that (—K, K)
contains the support of g. Extend g periodically, with period 2K, from (-K, K]
to the whole real line by setting g^K) (x) - g(x- 2Kj) if x € {2Kj - K, 2Kj +
K], j = ±1,±2,.... Since g^K) is a continuous periodic function whose real
and imaginary parts are, respectively, even and odd,
Re j 9{K)(x)Ms(dx) ~ Re f°° 9{K){x)MT(dx). (6.5.10)
J— CO J— OO
Since g is bounded,
IffM - g{K)(x)\am(dx)
i
r+oo
\g(x) - g^K){x)\al(-K,K}"(x)m(dx) -> 0
J —<
I
J —c
as K —* oo by the bounded convergence theorem. Applying Proposition 6.2.3,
we see that
/CO TOO
g{x)Ms{dx) = plim^-^^ Re / g{K){x)Ms{dx),
-oo J —oo
/co /*oo
g(x)MT(dx) = plim^^ Re / g{K){x)MT{dx).
-CO J —OO
From (6.5.10), we conclude that (6.5.9) holds for any continuous g with bounded
support whose real and imaginary parts are, respectively, even and odd.
Let A be a Borel subset of the real line. We want to show that
ks(Axds) = kT(Axds), (6.5.11)
where kT is the circular control measure of the SaS random measure MT. If
(6.5.11) holds, then we will be able to conclude Ms = MT.
Note thatit is enough to prove (6.5.11) for .A of the kind A = (a,6), 0 < a <b
or a < b < 0. Indeed, suppose (6.5.11) holds for every set A as above. Then, by

296 COMPLEX STABLE STOCHASTIC INTEGRALS 6.5
the monotone class theorem (6.5.11) holds for every Borel set A such that either
A C (0, oo) or A C (—oo, 0) and hence (6.5.11) follows from the decomposition
A = (,4n(-oo,0))U(J4n{0})U(An(0,oo)) and the assumption that m({0}) =
0. We will establish
ks((a,b) x ds) = kT((a,b) x ds) (6.5.12)
for every 0 < a < b (the case a < b < 0 is identical).
Let 6\ and 62 be arbitrary real numbers. Define
r{x) = 0l(l(a,6)(aO + l(-6,-a)(x)) + i02(l{-b,-a)(x) ~ l(a,6)0c))-
Let {/in}^L1 be a sequence of continuous real functions vanishing outside the
interval (a, b) such that hn(x) f 1 for every x € (a, b). Let
rn(x) = 9i(hn(x) + h„{-x)) + i92{hn(~x) - hn(x)), n= 1,2,....
Clearly, each rn is a continuous function with bounded support whose real and
imaginary parts are, respectively, even and odd. Applying (6.5.9), we have
/OO /»OC
rn(x)Ms(dx) = Re / rn(x)MT(dx), n = 1,2,....
-OO «* — OC
Moreover, Re J^or(a;)Mg(dx) = plim,,,,^ Ref*™ rn(x)Ms(dx) and
Re/^L r(x)MT(dx) = plim^^ Re J"^ rn(x)MT(dx), by Proposition 6.2.3.
Hence,
/oo /»oo
r(x)Ms(cte) = Re / r(x)MT(dx).
•00 j —00
But this last equation states that two (real) SaS random variables have the same
distribution. Applying Proposition 6.2.4, we have
/.
s\B\ + s2e2\a{ks + ks)((a,b) x ds)
Si
00
= / / \8\9i + s202\al(atb)(x)ks{dx,dB)
+ / / |si0,-s26i2ri(-b,-a)(x)fes(dx,ds)
J-00 -/S2
= / / |si Rer(x) — s2lmr(x)\aks{dx,ds) (definitionofr(x))
J—00./ S2
= / / \siRtr(x) — s2lmr(x)\akT(dx,ds) (Proposition 6.2.4)
J-00 J Si
= [ \8iei+s1B2\0'(kr + kr)((a,b)xds):
Js-,

6.5
HARMONIZABLE PROCESS
297
We can now appeal to Theorem 2.3.1 (uniqueness of the spectral measure of a
SaS random vector) and conclude
(ks + ks)({a,b) x ds) = (fcr + kT)({a,b) x ds).
(6.5.13)
Courage, tired reader! We are very close to the end of the proof (of Step 1)!
Relation (6.5.13) is very similar to Relation (6.5.12) which we want to establish.
Since
fci = (fc + k) = k + (k) = k + k = ks,
the left-hand sides of (6.5.13) and (6.5.12) are, in fact, identical. Thus, to establish
(6.5.12), it is sufficient to show that kT = kT.
Using the definition of kT, we obtain for any real 6\, 62 and any Borel set A
f \9iSi + 82s2\ak^{A x ds)
= / \6\S\ ~ 02s2\akT({~A) x ds)
Js2
— \ \ #i(si cosrx — S2sinTx) — 62{s\ sin tx + S2 cos ti) ks{dx,ds)
= / / 9\(s{COSt(-x) - (-s2)sinr(-x))
J A Js2
a
-92{si sinr(-x) + (-s2)cost(—x)) ks{dx,ds)
~ I I ^i(si costo; — sasinra:) +^2(^1 sinra; + S2Cosr:r) ks{dx,ds)
J a Js2
= / leis^^szrfcrlAxds),
JSi
where the second equality follows from Exercise 6.8, the fifth equality follows
because ks = ks, and the last equality follows once again from Exercise 6.8.
Applying Theorem 2.3.1 once more, we have ks{A x ds) = kT(A x ds). This
proves (6.5.12) and, therefore, completes the proof of Step 1 for the real-valued
harmonizable SaS process.
The argument for the complex-valued harmonizable process Z defined by
(6.5.1) is stationary is very similar and, in fact, simpler. The details are left to the
reader (see Exercise 6.13). This completes the proof of Step 1.
Proof of Step 2. We must prove that if MT = M for every real t, then
M = e'*M for every real <$>. In other words, we have to show that the circular
control measures of M and e^M coincide. Let k^ denote the circular control
measure of e^M. We must then prove that for every Borel set A, we have

298 COMPLEX STABLE STOCHASTIC INTEGRALS 6.5
k^(A x ds) = k(A x ds). By the monotone class theorem it is enough to prove
that for any real a < b, fc^((a, b] x ds) = k((a, b] x ds). Since m({a}) = 0, it
is sufficient to consider the case 0 < a < b. That is, we will prove
ei*M((a,&]) = M((a,6]) (6.5.14)
foranyO < a < b. LetA = b — a > O.andforann > 1,setck(n) — a+-A, k =
0,1,... ,n. Thus co(n) = a, Cn(n) = 6. Let
Jk(n)= f ei<t,x/Ck-'^M(dx), k=l,...,n,
and J(n) = YX=\ Jk(n). The key observation is that for each k = 1,..., n,
Jk(n) = M^/Ck_,(n) ((cfc_i(n),cfc(n)]) = M ((cfc-,(n),cfc(n)]).
Hence,
n n
J(n) = J2 J*(n) = EM ((c*-i(n),cfc(n)]) = M((a,6]),
fc=i fc=i
where the equality in distribution is between two sums of independent random
variables whose corresponding terms are equal in distribution. Therefore, for each
n = 1,2,..., the distribution of J(n) coincides with that of M((a, b]). In order
to establish Relation (6.5.14), it is now sufficient to prove
plim^ J(n) = e'*Af ((a, 6]). (6.5.15)
Note that
J(n) = f f^(x)M{dx) , n = 1,2,...,
J(a,b\
where
/„(*) = c'*1'0*-'^ if ck^(n) <x< ck(n), k = 1,... ,n.
By the Bounded Convergence Theorem, /(ob] |/„(z) - ei4,\am{dx) -+ 0 as
n -» oo and, hence, Proposition 6.2.3 yields Relation (6.5.15). This completes
the proof of Step 2 and also the proof of the necessity part of the theorem.
Proof of the sufficiency part of Theorem 6.5.1: Observe that (6.5.4) is equivalent
to
ks = ms7- (6.5.16)

6.5
HARMONIZABLE PROCESS
299
To prove sufficiency, we assume (6.5.16) and prove that the complex-valued
harmonizable process
/oo
eitxMs{dx), -oo < t < oo (6.5.17)
-oo
is stationary. This will establish the sufficiency part of the theorem for both the
real- and complex-valued cases.
Take any tx, t2,..., td € E and u > 0. By Proposition 6.2.1, for any real
number, (^.flf));^,^),...,^.^)..
d
Sexpji Yffi Re w$i + u) + e? Im Wfe + u))}
= exp{- £° J \fyisi8W + sjOf^cosixitj + «))
+(s!0$2) - 82^°) sinfx^ +u))|afcs(da;,ds)]}
. /-oo /• . d
= expj - / ms{dx) / s, XX'° c°s(a:(ij + u)) + of] sin(x(tj + u))]
J-co JS2' j=1
d
+s2 £[0$2) cos(*(*j + u)) - ^ sin(a:(tj + u))] "^(ds)}
c r°° r i d
= exp{-/ ms{dx) s, Re (^fljei(^-^+u)l>)
+ s2Im(^fljei(<k-^+u)l))| 7(ds)} (6.5.18)
where R^ = 6^ + ief1, j = 1,2,..., d. We now apply Proposition 2.5.5 to
conclude that the integral Js depends only on the magnitude
d d d
j=i j=i i=i
i.e., the characteristic function in (6.5.18) does not depend on u. This proves the
stationarity of the process W defined in (6.5.17), and thus the proof of the theorem
is complete. I

300 COMPLEX STABLE STOCHASTIC INTEGRALS 6.6
Example 6.5.2 The complex-valued harmonizable process considered in
Example 6.3.6, {H(t) — J^ eltxM(dx), t € R}, is stationary because the complex
random measure M in that example is isotropic.
Remark. In the proof of Theorem 6.5.1, we encountered a uniqueness
problem. Setting Xi(t) = /^oeitxM(rfx), -co < t < oo, and X2(t) =
/^o eltxMr (dx), -co < t < oo, we inquired whether X\ = X2 implies that M
and MT have the same circular control measure. To solve this problem, we used
a technique analogous to a standard one in the theory of characteristic functions
of probability laws, namely that two laws with identical characteristic functions
must coincide. (In our context, the "law" is the random measure M.) It is
possible, however, to use the analog of another standard technique for establishing the
uniqueness of characteristic functions of probability laws, that of inverse Fourier
transforms. We will see in Section 11.6 that there is an "inversion formula" for
harmonizable SaS processes, but we cannot use it at this point because it
involves integrating the sample paths of a-stable processes. This technique will be
developed in Chapter 11.
Proposition 6.5.3 A harmonizable SaS process is continuous in probability.
Proof: Since J^ \eltx — etsx|am(dx) —► 0 as s —* t, we conclude from
Proposition 6.2.3 that plims^tX(s) = X(t). I
6.6 Stationary real harmonizable processes
We consider in this section some consequences of Theorem 6.5.1 concerning
stationary real harmonizable processes.
Let M be a complex random measure with circular control measure k. If
{Z(t) = /^ eitxM(dx), t € M} is stationary, then its real part X = Re Z is
also stationary. The converse is not necessarily true. However, given a stationary
real harmonizable process X, there always exists some stationary complex-valued
harmonizable process Z such that X = Re Z. The proof of Theorem 6.5.1 shows
that this special Z can be chosen as 2~' la /^ eitxMs (dx) where Ms is isotropic
with circular control measure ks — k+k, where k is the conjugate measure denned
in (6.5.2). Indeed, X = Re Z stationary =*• ks = (m + rnj-y =►' Z stationary. By
(6.5.6), X = Re Z. Therefore,
Proposition 6.6.1 A real harmonizable SaS process X is stationary if and only if
there is some complex-valued harmonizable SaS process Z such that X = Re Z.

6.6
STATIONARY REAL HARMONIZABLE PROCESSES
301
Example 6.6.2 Fixuo > Oandletm = 6{_Uo}+6{uo}bethecontrolmeasureofa
complex-valued SaS random measure M. To define the circular control measure
k of M, we must specify fc{{—uq} x •} and k{{uo} x ■}. Set fc{{-uo} x •} =
k{{uo} x •} = 713, where 713 is the uniform probability measure concentrated on
the first and third quadrant of the circle 52. Then (6.5.3) does not hold because 713
is not the uniform probability measure on the whole of S2. Therefore the process
U^oo eltxM(dx), —00 < t < 00} is not stationary.
Now let 724 be the uniform probability measure concentrated on the second
and fourth quadrant of 52. Since k{{—u0} x •} = fc{{iio} x •} = 724, we obtain
k+k = (m+m)(713+724)/2where7 = (713+724)/2 is the uniform probability
measure on 52- Since (6.5.4) is satisfied, X(t) — Re J^ eitxM(dx) is stationary
with control measure m+m = 2m. Moreover, X(t) can be represented as X(t) =
Re /^ e%txMs{dx), where Ms has circular control measure ks = (m 4- m)^.
The measure Ms is isotropic and the process {J^°oaeitxMs{dx), t e M] is
stationary.
Proposition 6.6.3 A stationary real harmonizable process {X(t), —00 < t <
00} has finite-dimensional characteristic function
J3exp{i(0,X(i,) + • • • + 0dX{td))}
d , d
, r°°./" v2 ," \2,a/2 ->
= exp< — Co / (/. Ojcostjx) + ( )► 9jSmtjX) m(dx) > (6.6.1)
^-°° 5=1 j=i
I- r°° 1 d d \a/2 1
= exp|-co / 5Z52^jflfcCOs((tj-ifc)o;) m(dx)\, (6.6.2)
•• J-oo' „•_ 1 t._, '
I a/2
) j Oflk cos((tj - tk)x)
co = ^- [~* (cos <t>)ad<t> (6.6.3)
27T J0
and m is a finite measure. The measure m is the control measure of the isotropic
complex-valued measure M in the representation X(t) = Re J^ eltxM(dx).
Proof: See Relation (6.3.9). I
Let {X(t), -00 < t < 00} be a stationary real SaS harmonizable process.
Like any other SaS process which is continuous in probability, it is, conditionally,
a centered Gaussian process (Proposition 3.11.3). It is natural to ask whether it
is, conditionally, a stationary centered Gaussian process. This question is both
important and non-trivial. It is important because the structure of stationary
Gaussian processes has been thoroughly studied and if a stationary real 5a5

302 COMPLEX STABLE STOCHASTIC INTEGRALS 6.6
harmonizable process is conditionally stationary Gaussian, it can have a number
of important properties. The question is nontrivial because not every stationary
SaS process is conditionally stationary Gaussian. We know from Corollary
2.5.4 that every stationary sub-Gaussian SaS process is conditionally stationary
Gaussian but we will see later (Example 10.3.4) that there are stationary SaS
processes that are not, conditionally, stationary Gaussian.
Proposition 6.6.4 Let {X(t), -co < t < oo} be a stationary real SaS
harmonizable process. Then X is conditionally stationary centered Gaussian. In
fact,
{X{t), -oo < t < oo} = {{C^miE^Git), -oo < t < oo}
where
oo
G(t) = J2 r7I/a(GJ.° cos tZj + Gf sin tZj), -oo < t < oo, (6.6.4)
j=i
and ba and {Tj, Zj} are defined in Proposition 6.4.4. The process G is,
conditionally on the TjS and ZjS, a stationary centered Gaussian process with autoco-
variance function
oo
EG(t)G(s) = Y^TJ2/a cosZj(t - s). (6.6.5)
j=i
PROOF: The series representation follows from (6.4.8). Conditionally on the TjS
and ZjS, G is a centered Gaussian process with autocovariance function
oo
•^ EG{t)G{s) = Y2rJ2/a(costZJcossZJ + sintZisinsZj)
j=i
OO
= ^rj2"* co* Zj(t-a).
Thus, G is stationary. I
We have discussed in some detail two classes of stationary SaS processes: the
sub-Gaussian processes and the real harmonizable processes. These two classes
have certain similarities. For example, they are both conditionally stationary
Gaussian. Are these classes really different? If so, are they disjoint? It turns out
that they are different and "almost" disjoint. More precisely,

6.6
STATIONARY REAL HARMONIZABLE PROCESSES
303
Theorem 6.6.5 The only stationary SaS process which is both sub-Gaussian and
harmonizable is
X{t) - cA'/2(Gi cosut + G2 sinut), -oo < t < oo, (6.6.6)
where c>0isa real constant, A ~ 5Q/2((cos sf-)2/a, 1,0), u > 0, and Gx and
Gi are i.i.d. standard normal random variables independent of A.
PROOF: Note that the SaS process given in (6.6.6) is stationary sub-Gaussian
because it is of the form
X{t) = Al/2G{t), -oo<t< oo, (6.6.7)
where {G(t) = c(G\ cos ut + Gi sin ut), —oo < t < oo} is a stationary centered
Gaussian process with autocovariance function c2 cos ut. This implies
d d d >-
Eexpji^Xte)} = expj^-^lcl-l^^^flfccos^ - tk)u\° }
j=i i=i fc=i
(6.6.8)
(see Proposition 2.5.2.)
On the other hand, {X(t), —oo < t < oo} is also stationary harmonizable
because it can be represented as
/oo
eitxM{dx), -oo < t < oo, (6.6.9)
-oo
where M is an isotropic SaS random measure with control measure m =
2~Q/2|c|Qc^'<5u, where cq is the constant in (6.6.3) and 8U is the unit mass at
u. (Compare (6.6.8) with (6.6.2).)
This shows that the SaS process given in (6.6.6) is both sub-Gaussian and
harmonizable.
To prove the converse, suppose that a stationary SaS process X can be
represented (in distribution) both by (6.6.7), with a stationary centered Gaussian
process G and autocovariance function R, and by (6.6.9), with an isotropic SaS
random measure M with control measure m. Choose any t\ < t%, and let
A = h.-t\. Then
(i4,/2G(t,),i4,/2G(t2))=(Re / °° eihxM(dx), Re f° eu>xM{dx)).
(6.6.10)
Proposition 2.5.2 and Proposition 6.6.3 give the characteristic functions of the
SaS random vectors in the left- and right-hand sides of (6.6.10). Equating these

304
COMPLEX STABLE STOCHASTIC INTEGRALS
6.6
characteristic functions, we see that, for any real d\ and 92,
■1„ , -- ,A«/2
(-(i?(0)^ + 2R(A)exe2 + R(o)e22)y
/oo
\6\ + 26xe2 cos Ax + 6%\a/2m(dx).
-oo
Setting mi = (2Q/2.R(0)-Q/2co)m, we obtain
/+oo
(02 + 20,6>2 cos Ax + 6>!)Q/2m, (dx), (6.6.11)
■oo
where p = R(A)/R(0). Setting 0\ = 1 and 62 = 0 shows that m, is a probability
measure. Suppose 6\ and 62 are not both zero and let u = 26\62/(0\ + 02). Then
u takes values in [—1,1], and thus, for every u in this interval,
/oo
(l+ucosAx)a/2m,(dx). (6.6.12)
■oo
Note that (6.6.12) equates two functions of u defined on [—1,1]. It is easy
to check (see Exercise 6.14) that these functions are infinitely differentiable on
(-1,1); after successive differentiation, we obtain, for k = 0,1,2,....
/oo
(cos Ax)fc(l + ucosAx)(Q/2>-fcmi(dx).
•OO
Substituting u == 0 yields
/oo
(cosAx)femi(dx), k = 0,1,2,.... (6.6.13)
-oo
Note that (6.6.13) represents equality of all moments of two random variables:
one is the constant p, the other is cos AX where X is a random variable with
distribution mi. It is well known that if two bounded random variables have
identical moments, then they must have identical distributions (Exercise 6.15.)
Therefore cos AX = p with probability 1. This implies that the probability
measure mi and, therefore, the control measure m are concentrated on the set
Va = {x £ (-co,+oo) : cos Ax = p], (6.6.14)
where p = R(A)/R(0). Recall that the points ti and t2 were chosen arbitrarily, so
the control measure m must be concentrated on the set Va for any A > 0. Choosing,
for example, A = 1, \/2 and V3 shows that the measure m is concentrated on a set
{-u, u) with u > 0, i.e., m = a6u + b8-u for some a, b > 0 (see Exercise 6.16).
From (6.6.11), we conclude p = cosAu for every A and hence -^d = cos Au.
This proves that the process X must be representable in the form (6.6.6). The
proof of the theorem is complete. I

6.7
CODIFFERENCE FOR HARMONIZABLE PROCESSES
305
6.7 The codifference for stationary real harmonizable
processes
The codifference r(t) was defined in Section 4.7 (see also Section 2.10). For a
real SaS harmonizable process X, it equals
r{t) = 2\\Xmaa-\\X{t)-X(f))\\aa
= co{2m(R)- J |2-2costx|a/2m(cte)}
^ J — oo '
r A001 tx a i
= co\2m{R)-2a / sin— m(dx)\.
*• J-co' *■ '
Although it may not converge as t —» oo, its Gesaro mean ^ JQ r(t)dt converges
as T —* oo.
Proposition 6.7.1 For a real harmonizable SaS process
im — /
T
lirn^ ~ f r{t)dt = Co <2m(R) - (2a~ f (smu)adu j m(R\{0})|.
(6.7.1)
T/2U //mif is positive for 0 < a < 2.
PROOF: Let K(x) denote the integer part of x/(2ir) and observe that
limT^00K(T)/T=l/27r:
fT { i rT r°° \ tx\a 1
/ r{t)dt = co < 2m(R) - 2Q - / dt / sin y m(da;) > .
Now
rT coo /• 1 K(Ta;) c2wfe
T
I dt sin | m(d») = / ^ £ / sin2 ds + °(1)
^m(R\{0})±^(sini)Qds
= m(R\{0})- /" (sin s)ads
n Jo
as T —* oo, proving (6.7.1).
If 0 < a < 2, by Jensen's inequality,
a/2 /1 \ a/2
If If /If \ a/2 / 1 \
I ^ (sin s)« ds = - J (sin2 .)■/** < (- jf sin2 s ds) = (-)

306 COMPLEX STABLE STOCHASTIC INTEGRALS 6.8
so that
rT
i r
l™ ™ / T{t)<& > co{2m(R) - 2a/27n(M\{0})}
'—oo T J0
= co{2m({0}) + (2-2Q/2)m(K\{0})}>0 I
Now, lim 5; /„ r(i)dt > 0 implies that r(t) cannot converge to zero. Since
r(i) converges to zero for moving average processes (Theorem 4.7.3), we obtain
Theorem 6.7.2 Stationary SaS moving average processes are different from real
harmonizable SaS processes for any 0 < a < 2.
This theorem should be contrasted with the Gaussian case a — 2. In that case,
a stationary process may have both a moving average and a real harmonizable
representation.
Remark. When a = 2, the right-hand side of (6.7.1) equals 2com({0}). If
r(t) = Cov (X{t),X(0)) -+ 0, then m({0}) = 0.
6.8 Exercises
Exercise 6.1 Use Kolmogorov's existence theorem to show that there is a
stochastic process {(MW(A),MW(A)), A 6 So} such that for every AuA2,...,Ad
in £o and pairs of real numbers (0J0,6f]), (^'\ 0J2)),..., (6{dl\ 6^]), one has
d
Yj^M^AA+efM^Ai)) ~ SQ(a,0,0), (6.8.1)
where
JEJS2 j—l
Show that this process is independently scattered and cr-additive, and that for any
A e £0, M^(A) and M^(A) are jointly SaS with spectral measure k(A x •).
Hint: To prove (6.8.1), express A\ U • • • U Ad (and, hence, each Aj) as a union of
disjoint sets.
Exercise 6.2 Prove that a complex-valued random measure M satisfies (6.1.4) if
and only if its circular control measure has the form k = my, where m is the
control measure of M, and 7 is the uniform probability measure on 52.

6.8
EXERCISES
307
Exercise 6.3 Show that M^ and M^ are real-valued SaS random measures
with control measures given by (6.1.5).
Exercise 6.4 Show that a complex-valued SaS random measure M has
independent real and imaginary parts if and only its circular control measure k is
concentrated on the four "lines" E x {0,1}, E x {0,-1}, E x {1,0} and
£x {-1,0}.
Exercise 6.5 Prove that the only isotropic complex-valued SaS, 0 < a < 2,
random measure M which has independent real and imaginary parts is the trivial
measure M = 0.
Exercise 6.6 Let M be a complex-valued SaS random measure on {E,£) with
control measure m, and let /, g e La(E, £, m). Show that jE f(x)M(dx) and
fB g(x)M(dx) are independent iff fg = 0, m-a.e.
Exercise 6.7 Let M be a complex-valued SaS random measure on {E, £) with
a control measure m and let cf>: E —* C be a measurable function. Let £o,v> =
{A E £ : fA \ip(x)\am(dx) < oo}. Show that the stochastic process M^A) —
JA i{)(x)M(dx), A E £o,i>, is a complex-valued SaS random measure on (E, £).
What is the control measure of M^? Prove that for any measurable function
/ : E -> C such that / o ip e La(E, £,m) the integral JE /(x)M^(dx) is well
defined and equal in distribution to JE f(x)ip(x)M(dx).
Exercise 6.8 Under the assumptions of Exercise 6.7, show that the circular control
measure k^ of the SaS random measure My, where ip = ifrM + ty^ an(i
M^(A) = jAtp(x)M{dx), is
/ |0is, + 92s2\aki,{Axds)
= 11 Ms^Hx)-s2^2\x))+92(si^l\x) + s24>M(x))\ak(dx,ds)
Jajs1
for any A € £o,y and real ^i»#2-
Hint: Use (6.1.2) and (6.2.4). The intermediate equality is then
-InUexpWfl.MjV) + e2M^(A))}.
Exercise 6.9 Let M be a (complex-valued) SaS random measure on E whose
control measure m is concentrated at the origin. Let / : R -> C. Show that the
process
/oo
f{xt)M{dx), -co < t < oo,
is a constant SaS process, i.e., P(X(tx) = Jf (fe)) = 1 for every tut2 € R.

308 COMPLEX STABLE STOCHASTIC INTEGRALS 6.8
Exercise 6.10 Suppose that / and M in Theorem 6.4.2 are real-valued. What are
the choices for (SJ , SJ ) that reduce (6.4.2) to the real-valued series
representation given in Corollary 3.10.4.
Exercise 6.11 Let G\,Gz be independent standard normal random variables.
^en \tGi+Giy^' (Gi+Giyn) nas tne vector uniform distribution over the unit
circle S2 and is independent of G\ + G\.
Exercise 6.12 Prove (6.5.6).
Exercise 6.13 Give a proof of step 1 of the necessity part of Theorem 6.5.1 for
the case when the complex-valued harmonizable process Z defined by (6.5.1) is
stationary.
Hint: Apply the approximating procedure used in the text to the process Z instead
of the process Y. Conclude that the following counterpart of (6.5.9) holds:
/OO rOO
g(x)M(dx) = / g(x)MT(dx)
-00 J —00
for every continuous periodic function g : M —> C. Deduce that k(A x ds) —■
kT(A x ds) for any Borel subset A of the real line.
Exercise 6.14 Show that the functions in the left- and right-hand sides of (6.6.12)
have infinitely many derivatives at every point of (— 1,1). Show that the functions
in the right-hand side may be repeatedly differentiated inside the integral.
Exercise 6.15 Suppose that a sequence of real numbers m,k, k = 1,2,..., are
moments of a bounded random variable with distribution F. Then there is no
other distribution that has m*, k = 1,2,..., as moments.
Hint: If \X\ < C,then \mk\ < Ck. Use Carleman's condition (Feller (1971), VII
(3.14)).
Exercise 6.16 Show that the intersection of the sets V\, V^ and V^, defined in
(6.6.14), is either empty, or of the type {-x, x}, x > 0.

Chapter 7
Self-similar processes
Self-similar processes are invariant in distribution under judicious scaling of time
and space. They are important in probability because of their connection to limit
theorems (e.g., Lamperti (1962)) and they are of great interest in modeling. Some
aspects of self-similarity appear in geophysics, hydrology, turbulence, economics,
communications and in relation to "1//noises." Self-similar processes are also
considered in physics, particularly in connection to the so-called "renormalization
group theory" and "critical phenomena." Taqqu (1986) provides a bibliographical
guide to many applications.
The scaling coefficient or index of self-similarity is a non-negative number
denoted H. A process X = {X(t), t € R} is self-similar with index H if, for
any a > 0, the finite-dimensional distributions of {X(at), t € R} are the same as
those of {aHX(t), t e R}. The process is H-sssi if it is self-similar with index
H and has stationary increments.
Brownian motion, for example, is 1/2-sssi. Although the increments of Brown-
ian motion are independent, those of other iJ-sssi processes can display long-range
dependence or long memory. If the process is Gaussian, long-range dependence
manifests itself by the presence of cycles of any order and, ultimately, by a
spectral density that diverges at the origin like a power function. Equivalently, the
covariances decay very slowly to zero, also like a power function. Mandelbrot
and Wallis (1968) refer to this phenomenon as the Joseph effect, an allusion to
the biblical figure Joseph who was faced with long periods of plenty followed by
long periods of famine.
Although Gaussian processes can display long-range dependence, their
marginal distributions, being normal, concentrate their mass around the mean.
Q-stable distributions with 0 < a < 2, on the other hand, allow for much greater
variability. Mandelbrot refers to the variability of stable random variables as the

310 SELF-SIMILAR PROCESSES
Noah effect, an allusion, we may recall, to the biblical figure Noah who lived
through an exceptionally heavy flood. By considering in this chapter a-stable
self-similar processes with 0 < a < 2, we are, in fact, dealing with models that
can exhibit both the Joseph and the Noah effects.
In Section 7.1, we introduce general #-sssi processes (not necessarily
Gaussian or a-stable) and describe some of their basic properties. We show that the
existence of moments limits the possible values of H and, consequently, if an H-
sssi process is a-stable, 0 < a < 2, then its self-similarity index H is restricted
to the interval (0, 1/a) if a < 1 and to the interval (0,1] if a > 1.
The Gaussian case, because it is a paradigm, merits a separate section. For
a given H G (0,1), there is basically a single Gaussian #-sssi process, namely
fractional Brownian motion. We provide two equivalent representations for that
process, a "moving average" representation and a "harmonizable" one. The
increments of fractional Brownian motion are called fractional Gaussian noise. They
are stationary and if 1/2 < H < 1, they exhibit long-range dependence; that is,
their autocovariance function decreases slowly at large lags, like a power function,
and their spectral density increases like a power function at low frequencies and
explodes at the origin.
In Section 7.3, we describe the basic properties of a-stable if-sssi processes
with 0 < a < 2. Whereas fractional Brownian motion is the only i?-sssi
Gaussian process with 0 < H < 1, there are many different iJ-sssi processes
when 0 < a < 2. The most common one is the linear fractional stable motion
considered in Section 7.4. Its self-similarity index H belongs to the interval (0,1)
and excludes the value 1/a. The a-stable L6vy motion introduced in Section
7.5, on the other hand, has self-similarity index H = 1/a. It has independent
increments and is the stable counterpart to Brownian motion. It is the unique 1/a-
sssi process, if 0 < a < 1, but there exist others if 1 < a < 2. For example, if X
is a 1-stable random variable, then X(t) = tX is a 1-sssi process. If 1 < a < 2,
then the log-fractional stable motion investigated in Section 7.6 is also 1/a-sssi.
The real harmonizable fractional stable motion considered in Section 7.7 is yet
another if-sssi process. The kernel in its integral representation has the same form
as the kernel of the harmonizable representation of fractional Brownian motion.
Its properties, however, are quite different from those of the linear fractional stable
motion. The complex harmonizable fractional stable motion defined in Section 7.8
is a complex-valued a-stable, iJ-sssi process. Finally in Section 7.9 we describe
a large class of H-sssi processes that are subordinated to other processes, for
example, sub-Gaussian or sub-stable processes.
In modeling, it is the increments of if-sssi processes that are of interest. Any
if-sssi process {X(t), teE} induces a stationary sequence {Yj = X(j + 1) -
X(j), j e Z}. The sequence {Yj} corresponding to a linear fractional stable

7.1
SELF-SIMILARITY
311
motion is of particular interest because it displays both the "Joseph" and "Noah"
effects of Mandelbrot. It is called linear fractional stable noise and is studied
in Section 7.10. We analyze its asymptotic dependence structure by using the
codifference r(j). For most, but not all, values of a and H, t(j') decreases
as jaH~a for large j. This is analogous to the behavior of the autocovariance
function in the Gaussian case a = 2. Simulations of Gaussian and stable fractional
noises and motions are presented in Section 7.11.
In Section 7.12, we discuss autoregressive-moving average (ARMA)
sequences with stable innovations, and in Section 7.13, we extend to the stable
case the family of models known as fractional ARIMA. A fractional ARIMA
can also display long-range dependence, and because its definition involves more
parameters than the linear fractional stable noise, it allows for greater modeling
flexibility.
7.1 Self-similarity
Let T be either E, R+ = {t: t > 0} or {t: t > 0}.
Definition 7.1.1 The real-valued process {X(t), t € T} is self-similar with
index H > 0 (i/-ss) if for all a > 0, the finite-dimensional
distributions of {X(at),t € T} are identical to the finite-dimensional distributions of
{aHX{t), t e T}; i.e., if for any d > 1, U,t2,... ,td 6 T and any a > 0,
(X(at,), X{at2),...,X(atd))^{aHX(t1), aHX(t2),...,aHX(td)).
(7.1.1)
The complex-valued process {X(t), t € T} is self-similar with index H if for all
a > 0, the finite-dimensional distributions of {Re X(at), Im X(at), t € T} are
identical to those of {aH Re X{t), aH Im X{t), t € T}.
We shall always suppose H > 0.
Notation. Relation (7.1.1) will be expressed succinctly as follows:
(X(at), t € T} ^ {aHX{t), t € T}, (7.1.2)
Sometimes, however, when we consider a./tce<i t 6 T, we write X(at) = a X(t)
to mean that the random variables X(at) and aHX(t) have identical (one-
dimensional) distributions.
Example 7.1.2 Brownian motion {X(t), t € R} is a Gaussian process with
mean 0 and autocovariance function EX(t\)X(t2) = min(ti, t2). It is H-ss with
H = 1 /2 because for all a > 0, —oo < ti, *2 < oo,
EX{ati)X{at2) = min(at1,at2) = amin(ii,t2) = E(al/2X{U)){al/2X{t2)).

312
SELF-SIMILAR PROCESSES
7.1
Example 7.1.3 The SaS Levy motion, 0 < a < 2 is H-ss with H ~ 1/a G
(1/2, oo). This process was defined in Example 3.6.1 for T = [0, oo). (To define
it, for T = R, set X(t) = X{(t) for t > 0 and X(t) = X2(-t) for t < 0,
where {X\ (£), t > 0} and {Xj(t), t > 0} are independent copies of a 5a5 L6vy
motion). It is easy to verify that the SaS L6\y motion X = {X(t), t G T} is self-
similar with H = 1/a. Indeed, since X has independent increments, it is enough
to show that for all a > 0 and t,s G T, X(at) - X(as) = ax'a{X(t) - X(s)).
But this follows from X(t) - X(s) ~ Sa(\t - s\^a,0,0).
Remarks
1. Relation (7.1.1) states that a change of the time scale is equivalent to a change
in the state space scale. Note, however, that it is the finite-dimensional
distributions (the "odds") that are invariant under the transformation, not
necessarily the sample paths. Lamperti (1962) uses the term semi-stability
to describe the transformation in (7.1.1). The term self-similarity, coined
by Mandelbrot, is now standard. That term is also used in the context of the
scaling of non-random objects, such as fractals (Mandelbrot 1982).
2. If T = R, then for each fixed t ^ 0,
f tHX{l) if t > 0,
X(t)^|t|HX(signt)={
{ \t\HX{-l) if t < 0.
This useful relation does not typically hold in the sense of the finite-
dimensional distributions.
3. If {X{t), t G T} is H-ss, and 0 G T, then for each a > 0, X(0) =
X{a0) = aHX{0) and hence
JC(0)=0a.s. (7.1.3)
A non-degenerate H-ss process cannot be stationary because if it were, we
would have for any a > 0 and t > 0, X{t) = X(at) = aHX(t) and we would
obtain a contradiction because aHX(t) —► oo as a —» oo. There is, nevertheless,
an important correspondence between self-similar and stationary processes.
Proposition 7.1.4 If{X(t), 0 < t < oo} is H-ss, then
Y(t) = e~tHX{et) , -oo < t < oo,
is stationary. Conversely, if{Y{t), -oo < t < oo} is stationary, then
X(t) = tHY'(In*), 0<t<oo,
is H-ss.

7.1
SELF-SIMILARITY
313
PROOF: Let 0i,..., 0d be real numbers. If {X(t), 0 < t < oo} is H-ss, then
for any t\,..., td 6 R1 and h > 0,
j=i i=i
d
d
proving that {Y(t), -co < t < oo} is stationary.
Conversely, if {Y(t), -oo < t < co} is stationary, then for any £ i,... ,td > 0
and a > 0,
d d
]T OjXiatj) = ^2 8ja"t"Y (In o + In i,-)
j=i j=i
d
= ^flj-a^tfy (In t,-)
i=i
d
- ^e^xitj),
proving that {X(t), 0 < t < co} is F-ss. I
Example 7.1.5 Let {X(t), t > 0} be Brownian motion. It is thus Gaussian, has
zero mean and is H-ss with H = 1/2. Therefore, (Y(i) = e_t/2X(e'), -oo <
t < co} is a stationary Gaussian process with mean zero and autocovanance
function
EY{U)Y{t2) = e-^u+H)EX(eu)X{e^)
= e-5(t|+t2)min(e\et2)
i.e., {Y(i), -oo < t < oo} is an Omstein-Uhlenbeck process.
Proposition 7.1.4 shows that there are many different self-similar processes.
We shall consider here those that have stationary increments. They are of great
interest in applications because they give rise to stationary sequences with
remarkable features.

314 SELF-SIMILAR PROCESSES 7.1
Definition 7.1.6 A real-valued process {X(t), t £T} has stationary increments
if
{X(t + h)- X(/i), t £ T} = {X{t) - X(0), * € T}, for all heT.
A complex-valued process {X(i), t £ T} has stationary increments if, for all
/i€T,
{Re (X(i + h)- X{h)), Im (X(i + h) - X{h)), t £ T}
= {Re (X(t) - X(0)), Im (X(t) - X(0)), t £ T}.
Definition 7.1.7 The process {X(t), t £ T} is H-sssi if it is self-similar with
index H and has stationary increments.
Example 7.1.8 Brownian motion is if-sssi with if = 1/2. From the discussion
in Example 7.1.3, we conclude that a SaS Levy motion is JJ-sssi with H — 1/a.
Remark. The following is a simple but useful fact: if {X{t), t £ R} is H-sssi,
then fox fixed t,
X(-t) 4 -X(t) (7.1.4)
since X(0) = 0 and X{-t) = X{-t) - X(0) - X(0) - X{t) = -X(i).
The next proposition shows that the existence of moments limits the possible
values of H. It uses the following technical lemma:
Lemma 7.1.9 Let {X(t), t £ M} be H-sssi. Then for anys^Q and t ^ 0,
P(X(8) = 0,X(t)=0) = P(X(l)=0)
and
P{X{s) ? 0, X(t) # 0) = P(X(1) ^ 0).
Proof: Let p = P(X(1) = 0). By self-similarity, for any s ^ 0, we have
P(X(s) = 0) = P(X(sign s) = 0) = p (7.1.5)
and, by the stationarity of the increments, for any s ^ t,
P(X(«) = X(t)) = P(X(a - 0 = 0) = p. (7.1.6)

7.1
SELF-SIMILARITY
315
Moreover, for any t and u and any M > 0,
P(X(t) = X(u) ± 0)
< P(\X(t)\ > M, \X(u)\ > M) + P(0 < \X(t)\ < M, 0 < |X(u)| < M)
< P(\X(t)\>M) + P{0<\X(u)\<M)
= P{\X(t)\> M) + P(0 < \X(l)\ < Mu~H),
which is arbitrarily small for M and u sufficiently large. Thus,
lim P(X(t) = X(u) ^ 0) = 0. (7.1.7)
Since, moreover,
P(X(s) = X{u))
= P{X{s) = X{u) ^ 0) + P(X{s) = 0, X(u) = 0)
= P(X(s) = X(u) ± 0) + P{X(s) = 0) - P(X(s) ='0, X(u) ? 0),
we have by (7.1.5) and (7.1.6),
P(X{s) = 0, X(«) ^ 0) = P{X{s) = X(u) ^ 0). (7.1.8)
Considering now s, t and u, we have
P(X(s) = 0, X(i) ^ 0)
< P{X{s) = 0, X{u) ^ 0) + P{X{u) = 0, X(t) ± 0)
= P(X(s) = X{u) ± 0) + P(X(u) = X(t) ^ 0)
by (7.1.8). We now use (7.1.7) to conclude that, for any s and t,
P{X{s) = 0, X{t) ± 0) = 0. (7.1.9)
Therefore, for any non-zero s and any i,
Ppf(s) = X(i) = 0) = P(X(s) = 0) - P(X(s) = 0, X(t) yt 0)
= P
= P(X(1)=0),
by (7.1.5) and (7.1.9). This proves the first part of the lemma. From (7.1.5) and
(7.1.9), we see for any non-zero s and t,
P(X(s) * 0, X(t) ± 0) = P(X(s) + 0) - P(X(s) ^ 0, X.(t) = 0)
- 1-p
= P(X(1)^0).
This proves the second part of the lemma. I

,»I (, SELF-SIMILAR PROCESSES 7.1
Proposition 7.1.10 If{X(t), t G K} is H-sssi and
P(X(1) # 0) > 0, (7.1.10)
then the relation
E\X(\)\i < oo
implies
0< #< l/7 if 0<7< 1,
ami ;V implies
0<H < 1 // 7> 1.
Proof: Suppose first 0 < 7 < 1. Then (a + 6)7 < a7 + 67 for any a > 0, b > 0
and (a + b)i < a< + W if a > 0, b > 0. Thus, |X(2)|t < \X(2) - X(\)\i +
|X(1)|7, and this inequality is strict on the set A - {X(l) ^ 0, X(2) - X(l) ^
0}. Since Lemma 7.1.9 and Relation (7.1.10) imply P{A) > 0, we have
E\X(2)P<E\X(2)-X(l)p + E\X(l)\''. (7.1.11)
But X{2) - X{\) = X(l) - X{0) = X{\) by the stationarity of the increments
and self-similarity. Hence (7.1.11) reduces to E\X{2)\i < 2E\X{\)p. On the
other hand, by self-similarity, E\X(2)\~< = 2"^E\X\\)\i. Therefore, 2H^ < 2,
i.e., H < 1/7.
If 7 > l,then£'|X(l)|p < 00 for any p < 1 and hence 0 < H < 1/pforany
p < 1, implying 0 < H < 1. I
Remark. Condition (7.1.10) is very weak. It is clearly satisfied if {X{t), t G R}
is an Q-stable process 0 < a < 2 (not identically equal to zero).
Figure 7.1 displays the region of permissable values for (a, H). Since
testable processes, a < 2, have finite moments of order lower than a and Gaussian
processes have finite moments of any order, we have
Corollary 7.1.11 Let {X(t), t G R} be a non-degenerate a-stable, 0 < a < 2,
H-sssi process. Then
a < 1 =» 0 < H < 1/a,
a> 1 => 0<#< 1.

7.1
SELF-SIMILARITY
317
H
H = \/a
H=\
0
1 2
Figure 7.1: If the process is, non-degenerate,
Q-stable and if-sssi, then a and H must lie in
the shaded region.
The corollary is consistent with Example 7.1.3 where we noted that the SaS
L6vy motion is H-ss with H = I/a. We will see in Section 12.4 that there is an
a-stable ff-sssi process for each permissible value of the pair (H, a).
Are there "different" a-stable if-sssi processes with a given a and HI In
order to eliminate trivial answers, we adopt the following convention:
Convention. Stochastic processes that differ only by a multiplicative constant are
identified. Hence, we say that X, = {X,(t), t S T} and X2 = {X2(t), t G T}
are different if there is no constant C such that X\ and CX2 have identical finite-
dimensional distributions.
We will see later that for fixed a 6 (0,2) and H, there are often many different
a-stable H-sssi processes. In the Gaussian case a = 2, however, there is a unique
if-sssi process, calledfractional Brownian motion. Fractional Brownian motion is
important not only because it is the unique H-sssi Gaussian process or because the
various a-stable H-sssi processes reduce to it when setting a = 2 but also because
it has been widely applied, particularly in the context of long-range dependence.
We shall now define it and study its properties.

318
SELF-SIMILAR PROCESSES
7.2
7.2 Fractional Brownian motion
A mean zero Gaussian random variable X has characteristic function Eeiex =
exp{-cr202} = expf-jcr2^2}, i.e., scale parameter a > 0 and standard
deviation <7n = \/lo-} The finite-dimensional distributions of a Gaussian process
{X(i), i 6 R} satisfy
£e^,<^> = exp{-i£ SA(tilt4)«iJi + E^J}1
i=i fe=i j=i
#,,..., 0m € 1, m > 1, where /x(t), ( 6 R, is a real-valued function
and {A(t],t2), t\,t2 € R} is non-negative definite; i.e., for any integer
m, ii,..., tm € R, ui,..., um G R,
m m
^^^(tj-.tj-JuiUj > 0.
t=i j=i
Conversely, to each /i and .A corresponds a Gaussian process: /x is its mean value
function and A is its autocovariance function.
FixnowO < H < 1. Since the function {|ti|2K + \t2\2H-\U ^-t2\2H, *i,t2-€
K} is non-negative definite (see Lemma 2.10.8), there exists a Gaussian process
{X(t), t € R} with mean zero and autocovariance function
RH(U,t2) := Cov(X(t,),^(t2))
= 5{|ii|2H + |t2|2H-|t. -t2|2W}varX(l). (7.2.1)
It is easy to check that this process is if-sssi. It is called fractional Brownian
motion.2
Are there other Gaussian iJ-sssi processes? The following lemma implies that
the answer is negative when H < 1.
Lemma 7.2.1 Suppose that {X(t), (€1} is a (non-degenerate) H-sssi finite
variance process. Then
0<H < 1,
X(0) = 0 a.s.,
'There is an inconsistency in the literature between the definitions of the scale parameter for the
Gaussian distribution (a = 2) and for the stable non-Gaussian ones, 0 < a < 2. If X ~ SaS
with 0 < a < 2, then the scale parameter of X is defined as the coefficient a in Eexp{i9X} =
txp{-cra\S\a}. If X is Gaussian, however, then its scale parameter is usually chosen to be the
standard deviation cto- m order to be consistent, we define the scale parameter as the factor a in
.Eexp{iflX} = exp{-CTa|0|Q} in all cases 0 < a < 2 and hence distinguish between a and the
standard deviation an — v2& in the case a = 2.
2The formal definition of fractional Brownian motion is given below.

7.2
FRACTIONAL BROWNIAN MOTION
319
and
Cov (X(U),X(t2)) = i{|i,|^ + \t2\2H - |t, - i2|2*} VarX(l).
Moreover, for all t € K,
(0 in the case 0 < H < 1:
£X(*) = 0;
(h) //I f/ze case if = 1 :
X(i) = tX{\) a.s.
PROOF: By the stationarity of the increments and self-similarity,
EX(U)X(t2) = l-{EX1{tx) + EX2{t2)-E{X{U)-X(t2))2]
= ^{£^2(r,) + £X2(t2) - E{X{t{ - t2) - X(0))2}
= \{\tA1H + \t2\2H -\tx-t2\w)EX\\) (7.2.2)
since X(0) = 0 by (7.1.3), X{\) = -X(-l) by (7.1.4) and X(t) =
\t\HX(sign t) for fixed t. Corollary 7.1.11 and Lemma 2.10.8 yield 0 < H < 1.
Suppose,now.O < H < 1. Since £?X(1) = E(X{2)-X(l)) = 2HEX{1)-
EX{\) = (2H - l)EX{t), we obtain EX(1) = 0 and hence EX{t) = 0,
because EX{-\) = -£X(1) and £X(t) = |i|H.EX(sign i). Form (7.2.1) of
the autocovariance function follows from (7.2.2).
Suppose now H = 1. Then (7.2.2) yields EX{U)X(t2) = txt2EX2{\) and
thus
E{X{t)-tX{\))2 = EX2(t)-2tuE;X(r.)X(l) + t2.E;X2(l)
= {t2-2t-t + t2)EX2{l)
= 0,
that is, X(t) = rX(l) a.s. for all t. Form (7.2.1) of the autocovariance function
now follows. I
Lemma 7.2.1 implies that all ff-sssi Gaussian processes have the
autocovariance function (7.2.1). For a given H, these processes only differ by a
multiplicative constant (and by their mean, if H = 1). It is convenient, therefore, to define
fractional Brownian motion as follows:

320 SELF-SIMILAR PROCESSES 7.2
Definition 7.2.2 A Gaussian if-sssi process, 0 < H < 1, is called fractional
Brownian motion (FBM) and is denoted {BH(t), t € R}. It is called standard
fractional Brownian motion if a\ = Var X( 1) = 1.
There are also non-Gaussian finite variance i/-sssi processes (see Example
7.9.6; see also Taqqu (1981, 1988) and Major (1981)) but, here, we consider
only the Gaussian ones. The following corollary is an immediate consequence of
Definition 7.2.2 and Lemma 7.2.1.
Corollary 7.2.3 FixO < H < 1 and a} = EX2 (I). The following statements
are equivalent:
(i) {X(t), t € T} is Gaussian and H-sssi.
(ii) {X(t), t € T} is fractional Brownian motion with self-similarity index H.
(Hi) {X(t), t € T} is Gaussian, has mean zero (if H < 1) and autocovariance
function (7.2.1).
Example 7.2.4 When H = 1/2, fractional Brownian motion {BH(t), t S K} is
Brownian motion because (7.2.1) reduces to
{ctq min(ti, t2) if t\ and ti have the same signs,
0 if ti and ti have opposite signs.
Example 7.2.5 For H = 1, we have B\(t) = t(Z + fx) by Lemma 7.2.1, where
H € R and Z is a mean zero normal random variable.
Because the case H — 1 is degenerate, we shall suppose in the sequel that
0<H < 1.
Defining a process through its finite-dimensional distributions does not always
give a clear picture of its underlying structure. An integral representation is
often more helpful. The following subsections present the most common integral
representations of fractional Brownian motion. The increment process, called
fractional Gaussian noise, is introduced in the final subsection.
7.2.1 "Moving average" representations of fractional
Brownian motion
We want to obtain integral representations for fractional Brownian motion of the
form JZo ft(x)M(dx).
Let (E,£,m) be a measure space and M a SaS random measure, 0 <
a < 2, with control measure m. The definition of the stable integrals I(ft) =

7.2
FRACTIONAL BROWNIAN MOTION
321
JE ft(x)M{dx), t <E T, given in Section 3.4 applies to any 0 < q < 2. When
a = 2, 7(/t) is well defined if JE \ft(x)\2m(dx) < oo and one has
EI(ftt)I(ft2) = Cov (J(/t|),!(/«,)) = 2 / /tl(a;)/t2(x)m(dx)
by (3.5.4). In particular, EM2(A) = 2m(A) for any A e £ with finite measure
m.
Fix a = 2 and let
(E,£,m) = (R,B,\-\/2),
where B is the Borel cr-field and | ■ | is Lebesgue measure on E. Then EM2 (A) =
\A | for any A S B with a finite Lebesgue measure and
/oo
ftl(x)ft2(x)dx. (7.2.3)
-oo
The following proposition gives a first integral representation for fractional Brow-
nian motion. We shall use the function
( u ifu>0,
u+ = <
[ 0 ifu<0,
and adopt the convention 0° = 0.
Proposition 7.2.6 Let 0 < H < 1. Then standard fractional Brownian motion
{Btf (£), t 6 R} /za.y the integral representation
^^/"(((t-x^-'^-a-^+^-^MCdx), t€R, (7.2.4)
vw/iere
C,(ff) = {|°° ((1 + *)*-'/2 - x^'/2)2^ + ^}'/2. (7.2.5)
VWzen If = 1/2, C\{l/2) = 1 and the representation (7.2.4) is to be
interpreted as /„' M{dx) ift>0and- j° M{dx) ift<0, i.e., as an integral
representation of Brownian motion.
PROOF: Let X(t) denote the integral in (7.2.4) and let ft(x) denote its integrand.
In order to verify that X(t) is well defined, we show firstly that /^ }2[x)dx <
oo. This relation is obvious when H - 1/2, because /^ f2{x)dx = /„ da; <
oo. Suppose now 0 < H < 1, H ^ 1/2. Then as x -* -oo, ft{x) ~ (# -
l/2)(-i)H_3/'2 whose square is integrable around -oo, /t(x) ~ (t - x)+

322 SELF-SIMILAR PROCESSES 7.2
as x —> t whose square is integrable around x = t, and similarly for x = 0 and
x — oo. Hence f^ f2(x)dx < oo and (7.2.4) is well denned.
We now verify that {X(t), t £ R} has the autocovariance function (7.2.1)
with VarX(l) = 1. Notice that X(0) = 0 a.s. and, for every t > 0, JSX2(t)
equals
_^I£((«-„?-/»-(t,,;-^(b
i r°°
1 *2ff / //, „ \H-l/2 / ^-1/2n2j
= ^f< y_J(1_u)+ ~("u)+ )du
/° ((i-«)ff-,/2-(-U)a-»/2)2d«+ Ai-t*)2"-^!
J-oo 7o -I
C,(ff)
1 t2H
1 t2K
\{l+x)M-M2_xH-M2)2dx+±.
= t
2H
Cx{Hf
and similarly EX2[t) = \t\2H for t < 0. Further, for any t,s S M,
£(X(t)-X(s))2 = ^j—jjit-x)«-l'1-{8-x)»-1'2)'
= 1 /"°°f(t-s-i)H-1/2-(-x)K"1/2V
by the previous calculation and hence (7.2.1) follows. I
The kernel
dx
dx
Mx) = ((t-x)+)H-^2-((-x)+)H-^2
(7.2.6)
is plotted in Figures 7.2 and 7.3 for several values of t and several values of
d = H-1/2.
The representation (7.2.4) is far from unique. In fact, let
0 ifu>0,
—u if u < 0.
Then for any real a and b numbers,
J—oo L
+ b{(t-x)"-l/2-(-x)"~l/2}]M(dx), t€R, (7.2.7)

7.2
FRACTIONAL BROV/NIAN MOTION
323
t = 10
d = 0.2
2-
Figure 7.2: Kernel (7.2.6) of fractional Brownian motion {d = H - 1/2) and kernel
(7.4.2) of linear fractional stable motion {d = H - 1/a, a = 1, b = 0), with rf = 0.2
and d = —0.2, plotted for several values of t.

324
SELF-SIMILAR PROCESSES
- 7.2
Figure 7.3: Kernel (7.2.6) of fractional Brownian motion (d = H — 1/2) and kernel
(7.4.2) of linear fractional stable motion (d = H — 1/q, o = 1, 6 = 0), with t = 1,
plotted for several values of d.

7.2
FRACTIONAL BROWNIAN MOTION
325
is also (up to a multiplicative constant) a representation of fractional Brownian
motion (Exercise 7.2).
The representation of fractional Brownian motion given in Proposition 7.2.6
corresponds to a = \/C\{H) and 6 = 0. That representation is "non-anticipative"
because it involves only integration on a; € (-oo,i]. Whena = b = 1,then (7.2.7)
yields the "well-balanced" representation
/oo
(\t-x\H-l^-\x\H-l^)M{dx).
-CO
7.2.2 "Harmonizable" representations of fractional Brownian
motion
We now turn to representations of the "harmonizable" type, i.e., of the form
I (ft) — J-oc ft(x)M(dx) where ft is complex and M is a complex isotropic
random measure. Integrals ot his type were defined in Section 6.3 for the a-stable
case, 0 < a < 2, but these definitions apply to the Gaussian case as well.3 If
we adopt this approach, J^ ft{x)M(dx) will be,.in general, complex and we
will have to consider Re J^ ft{x)M(dx), for example, if we want a real-valued
process.4
Here, we present an alternative definition, one which has been traditionally
followed in the context of Gaussian self-similar processes. In this approach, the
real and imaginary components M^ and M^> of M are independent but they are
not independently scattered, i.e., M^\A) may not be independent of M^(B)
even for disjoint sets A and B. This second approach allows us to choose the
complex ft and M in such a way that the integral J^ ft(x)M(dx) is always
real-valued.
Firstly, some heuristics. We would like to view the complex-valued ft and Mt
as "Fourier transforms" of real-valued ft and Mt. We want them to be defined in
such a way that a Parseval-type relation /^ ft(x)M(dx) = /^ ft(x)M(dx)
holds. Since ft and M are real, ft and Mt should satisfy ft{x) = ft(~x)
and M(A) — M(—A), where a bar denotes a "complex conjugate." The relation
M(A) = M(-A) implies that M(A) and M(—A) are dependent even for disjoint
sets A and — A; hence, the complex-valued random measure M will not be
independently scattered on all Borel subsets of R.
3 A special feature of the Gaussian case is that the real and imaginary parts of M are independent.
4The reader who prefers this approach will find it in Section 7.7 where we define the real
harmonizable fractional stable motion for 0 < a < 2. It is an harmonizable representation for fractional
Brownian motion when a = 2.

326 SELF-SIMILAR PROCESSES 7.2
The following definitions of the random measure M and the integral
/(/) = J^ f(x)M(dx) are motivated by the preceding heuristics. Define
firstly M — M^ + iM^ where M'1' and M^ are independent Gaussian
random measures, independently scattered on R+ and satisfying M^(A) =
MM(-A), MW{A) = -MW(-A) for any Borel set A of finite Lebesgue
measure. Then M(A) = M(—A). Suppose also that M'1' and M^2) have control
measure (1/4)| -Jjso that on R+, E(M^(A))2 = £(M<2>(A))2 = i|yi|. Then
the variance of M(A) equals
£|M(,4)|2 = E(M^(A))2 + E(M<V(A))2 = \A\
for any Borel set A € R+. Since M is complex, the quantity E(M(A))2 has no
special significance. In fact,
E(M(A))2 = E(M<'>(yi))2 - £(M<2>(,4))2 = 0.
We now describe the integrands. Let / = /M + if^ where /^ and /(2) are
real-valued functions that satisfy f^l)(x) = f^(-x), f^(x) = — f^(—x) and
f^if^Hx^dx < oo. Thus, /belongs to the set
-F={/:/(x) = /(-x) and / |/(x)|2dx < oo }.
For any / e JF, the integral
/oo __^
/(x)M(dx)
-oo
is defined as
,-0
/(x)M(dx) +
Jo
(/(1)(x) - i/(2)(x))(M(1)(dx) - iM^idx))
i.e.,
/(x)M(dx) + / J{x)M{dx)
•oo JO
Jo
/•OO
+ / (/0)(z) + if{2)(x))(M^(dx) + iAf^(di))
Jo
= 2(f°/(1>(x)M(1>(dx)- [°° fW(x)Mm(dxj), (7.2.8)
/oo /»oo
f^(dx)M^(dx)- / /^(x)M^(dx).
■oo J -~<x>

7.2
FRACTIONAL BROWNIAN MOTION
327
{ J(/)> / 6 F} is a Gaussian system in which each /(/) is a real-valued Gaussian
random variable with mean zero and variance
xdx\
oo.
£(|"/(x)M(dx))2 = ^jfVWy + jTVW-
= 2 / |/(a;)|2dx = / \f(x)\2dx <
Moreover, the covariance of I(f\) and /(/a), /i, h € J7, equals
/oo ■
fl(x)f2(x)dx. (7.2.9)
-oo
With the preceding definitions, we can now start with a set of integrands
Fo = {ft- / \ft(x)\2dx < oo, /t real-valued, t £ t\,
and define for each ft € ^o its L2 Fourier transform ft which we denote (with
some abuse of notation) by
/t(*) = 4= fX e^ft(u)du.
We shall now try to relate the Gaussian processes {/(/t) = f^ooft(x)M(dx), t €
T} and {I(ft) = J^ ft(x)M(dx), t e T}. Here, M is a real-valued Gaussian
random measure and M is the complex Gaussian random measure defined above.
Observe that, since ft 6 Jo is real-valued, its Fourier transform ft satisfies
ft(x) = ft(-x).
Proposition 7.2.7
ft(x)M(dx), teT)± {l{ft) = J_Jt(x)M(dx), t e T}.
Proof: Since both {/(/*), t € T} and {/(/t), t e T} are Gaussian with mean
zero, it is sufficient to prove that for any two functions ft), ft2 6 -^o one has
EI(fu)I(ft2) = EI(Jtl)I{ft2). This, however, is an immediate consequence of
(7.2.3), (7.2.9) and the Parseval identity:
/t,(s)/t2(s)di= / fu{x)ft2{x)dx. ■
-co J—oo

328 SELF-SIMILAR PROCESSES 7.2
Thus, in order to obtain a harmonizable representation {J^°ooft(x)M(dx),
t £ 1} of fractional Brownian motion, one can start with any one of its
representations {/^j ft(x)M(dx), t € R}, and derive the Fourier transform ft of ft.
This can be delicate at times and therefore it is easier to first guess the form of ft
and then verify that the resulting process {I{ft), t € E} is i?-sssi.
Consider, for example, the representation (7.2.4) of fractional Brownian
motion. A heuristic computation5 suggests the following representation (7.2.12).
Proposition 7.2.8 Let 0 < H < 1. Then standard fractional Brownian motion
{£?#(£), t € K} has the integral representation
/oo Ami _ i
: \x\-(H~WM{dx), teR, (7.2.12)
-oo lx
C2(H)
where
a""-(ar(2^,J,)"i' <7-2-'3>
PROOF: Let X{t) denote the integral in (7.2.12) and Jt{x) its integrand. Since
|(el2t - l)/ix\ is bounded for x —► 0 and behaves like |x|~' as x —► ±oo,
one obtains j"^ \ft(x)\2dx < oo. Hence, X(t) is well defined. It is easy to
verify that the process is H-sssi (Exercise 7.6) and therefore by Corollary 7.2.3,
{X(t), t e R} is fractional Brownian motion.
5Here, we present the heuristics leading to (7.2.12). The computations are not precise because the
integrals diverge! Suppose H > 1/2 and t > 0 and rewrite (7.2.4) as
cr1
(H)(H - 1/2) J ( M« - 0+"3/2ds)M(dO (7.2.10)
= C;l(H){H - 1/2) f if \m(s)(s-0+'V2ds\M(d4). (7.2.11
J —oo J — oo
)
The inner integral in (7.2.11) is the convolution of the function 1 |0ti (s) with s, _ and its Fourier
transform should equal the product of the convolutions. Now the Fourier transform of 110 tj (s) is
\—1 /^ixt
(VSrT1 f eizsds = (V2^r'(ix)-]{ei-
Jo
!)•
Although the Fourier transform of s+ does not exist (the function is not in L1 nor X,2), let us
write, nevertheless,
./-oo -/O
by setting u = xs. Viewing the last integral as an improper Riemann integral, one can show that it
equals exp{iir7/2}r(H — 3/2) (see Exercise 7.4). Use (/ x g) = V2wJ§ and Proposition 7.2.7 to
obtain (7.2.12).
This heuristic can be made rigorous (Exercise 7.5).

7.2
FRACTIONAL BROWNIAN MOTION
329
It remains to show that EX (I)2 = 1, i.e.
/oo ^
\h{x)\2dx = C2{H). (7.2.14)
-oo
Since \ezx - \\2 — (cos a: - l)2 + sin2x = 2(1 -cosz) = 4sin2 f, we have
£ \Jx{x)\2dx = &j°° S-^^dx. (7.2.15)
If H = 1/2, this equals 8?r/4 = 2tt = C|(l/2) using Relation GR3.821 in
Gradshteyn and Ryzhik (1980).
If if + 1/2, use Relation GR3.823 in Gradshteyn and Ryzhik (1980), 2r(z) =
T(z + 1) and T{z)T{l - z) = ir/ sinirz to show that (7.2.15) equals
8 2r(2-2#)cos#7r 2
--r(-27r)cosffT = g(2J? _ 1} = C2{H).
(T{z) is the Gamma function.) This proves (7.2.14) and EX(l)2 = 1.1
Using (7.2.8), we can write the representation (7.2.12) as
2
C2(H) .Jo
I r !^x-(H-./2)M(.)(da;
+ ri-c™xtxHH-u2)M(2){dx)\ (7216)
Jo x >
where M^ and M^ are two independent real-valued Gaussian random measures
such that E{M^{A))2 = \A\/2 for any Borel set A of E+. The integrands of
M(1) and M(2) are plotted in Figures 7.4 and 7.5, for t = 1 and d = H - 1/2 =
±0.2.
Remarks
1. When H = 1/2, C2(l/2) = Vllr and (7.2.12) gives the following
representation for Brownian motion,
t r°° pixi — 1 ^
|B(t) = V2^/ Af(dg), te»).
2. For any real a and 6, the process
| J" ilz±(a{x+)-^-in) +b(x_r(H-W)M(dx)} t6RJ,
(7.2.17)
is also fractional Brownian motion (up to a multiplicative constant) because
it is a Gaussian F-sssi process (Exercise 7.7). The representation (7.2.12)
corresponds to the choice a = b = C^x (H).

330
SELF-SIMILAR PROCESSES
7.2
(s\nxt)x d '
1
~4 x
(1 -cosxt):c_d_1
1
0.5
-0.5
d = 0.2, t = 1
^x.
5 10 15 20 25 30
Figure 7.4: Integrands of M(1) and M(2), with t = 1, in the representation (7.2.16) of
fractional Brownian motion (d — H — 1/2) and in the representation (7.7.4) of the real
harmonizable fractional stable motion (d = H — \/a). Here, d = 0.2.

7.2
FRACTIONAL BROWNIAN MOTION
331
d = -0.2, t = 1
(1 -cosxt)x-d-'
1
-0.5 -
Figure 7.5: Integrands of M(1) and M(2), with t = 1, in the representation (7.2.16) of
fractional Brownian motion {d = H ~ 1/2) and in the representation (7.7.4) of the real
harmonizable fractional stable motion (d = H - 1/a). Here, d = -0.2.

332
SELF-SIMILAR PROCESSES
7.2
3. The representation (7.2.12) (or (7.2.17)) has a physical interpretation. The
factor (elxt — \)/ix is the Fourier transform of the indicator function of
the interval [0, t) and ensures stationarity of the increments. The term
|a;|-(^-i/2) colors the underlying Gaussian noise. In a sense that will be
made precise in the following section, it gives rise to a spectral density
||x|-(#-i/2)|2 _ x-(2H-\)^ wrijcri diverges at iow frequencies (x —» 0)
when 1/2 < H < 1, is colorless (equal to 1) when H = 1/2 and tends to 0
at low frequencies when 0 < H < 1/2.
7.2.3 Fractional Gaussian noise
Since fractional Brownian motion {Bn(t), t € R} has stationary increments, its
increments
Yj = B„(j + 1) - B„(j), j = ..., -1,0,1,...,
form a stationary sequence. The sequence {Yj, j £ Z} is called fractional
Gaussian noise (FGN).6 It is called standard fractional Gaussian noise if erg =
Vary/=1.
Any representation of {.B#(t), t € K} induces a representation for {Yj, j g
Z}. For example, using (7.2.4) and (7.2.12) respectively, we obtain the moving
average representation
Yj ± ^y J3^ (<j + l-x)Z-l/2-U-x)"-l/2)M{dx), j 6 Z, (7.2.18)
and the harmonizable representation
yi = T^Tm [°° e^e^±\x\-(»-^M(dx), j £ Z. (7.2.19)
w(-"J J-oo lx
Fractional Gaussian noise has some remarkable properties. It is a stationary
Gaussian sequence with mean zero and variance EYf — EB2H{\) = o\. Let
{r(j) = EYqYj, j € Z}, denote its autocovariance function and let {/i(A), —it <
A < 7r} denote its spectral density, i.e.,
r(j) = ^ eiX^h(X)dX, j - 1,2,... . (7.2.20)
J — TV
6One sometimes defines fractional Gaussian noise as
Y(u) = BH(u+ I) - BH(u), ueR.
For most time series applications, however, one restricts u to Z.

7.2
FRACTIONAL BROWN1AN MOTION
333
(It is possible to obtain h from r by using the relation
j = — CO
whenever the sum converges.) The correlation and spectral density are plotted in
Figures 7.6 and 7.7 respectively.
Proposition 7.2.9 Fractional Gaussian noise has autocovariance function
r(j) = ^(\j+i\2H-2\j\2H + \j~l\2H), jeZ, {7.2.21)
and spectral density
2 oo
cm1
k= — oo
|A + 27TfcP"+'
, -7T < A < 7T. (7.2.22)
Equivalently,
,/,x <7? fn°°cosa;A(sin2 f)x 2H_1cfa:
MA = °Joroc, . ' 2' ,. , -7r<A<7r. 7.2.23
J0 (sin2 |)x_2H_1dx
PROOF: Assume, without loss of generality, that ctq = EYf — 1. Then
r(j) = £y0^ = SB«(1)(BhO'+1)--BhO'))
- RH(l,j + l)-RH{hj)
= ^{|j + l|2H-2|j|2H + |j-l|2H} (7.2.24)
by (7.2.1). An easy way to obtain the spectral density /i(A) is to evaluate the
autocovariance r(j) by using the representation (7.2.19) of {Yj, j € Z} and
Relation (7.2.9):
r(j) = EYoYj
■oo
■2 / tn / „i:rj
e,x - 1
■oo
oo rn+2-rrk
x\-HH-\ft)dX (7.2.25)
t _w J —ir-{-Ink
fc——oo
OO -7T
OO fit
= C2-2(H) E / e^'|ea-l|2|A + 27rfc|
k~ —oo
i-ir °° 1
-2H-'dA
dA,

334 SELF-SIMILAR PROCESSES 7.2
Figure 7.6: The correlation r(x) = (|x + 1
)/2 of fractional
Gaussian noise (see Relation (7.2.24)), plotted for several values of H. Note that it tends
much faster to zero as x —+ oo when H < 1/2 than when H > 1/2.
0<H<
2 " 2
Figure 7.7: Spectral density h(X) of fractional
Gaussian noise for H = 0.7 and H = 0.3 (see
Relation (7.2.22)). /i(0) is infinite if H > 1/2 and
is zero if H < 1/2.

7.2 FRACTIONAL BROWNIAN MOTION 335
and then deduce (7.2.22) by comparing this last expression with (7.2.20). To
obtain (7.2.23), use |eix - 1|2 = 4sin2 f, (7.2.14) and (7.2.15) in (7.2.25). I
If H = 1/2, then r(j) = 0 for j ^ 0 and hence {Yj, j = ..., -1,0,1,...}
is a sequence of i.i.d. Gaussian random variables. The i^s are dependent when
H ^ 1/2. The following result specifies the behavior of the autocovariance
function r(j) at large lags j, and, correspondingly, the behavior of the spectral
density h(X) at low frequencies A.
Proposition 7.2.10 Let {Yj, j = ..., -1,0,1,...} be fractional Gaussian noise.
Then
r(j) ~ a\H{2H - l)j2H~2 as j -> oo (7.2.26)
for H ± 1/2, and
h{X) ~ o-lCz2(H)\\\l-2H as X -» 0. (7.2.27)
Proof: For large j,
^-^(y[(. + if-2+(l-I)"]).
and the content of { } tends to 2H(2H — 1) as j —> oo. As for /i,
i °° 1
h(X) = 4C-(H)ie- - H^p^T + tE |A + 27rfc|2H+1}-
The result follows because |eiA - 112 ~ | A|2 as A -» 0, and the sum in the brackets
is bounded for all —n < X < ir. M
The following are important points:
• Both r(j) as j —► oo and h{X) as |A| —> 0 behave like power functions.
• r{j) tends to 0 as j —> oo for all 0 < H < 1, but when 1/2 < H < 1 it
tends to zero so slowly that Y°jL-<x> T(J) diverges- We say that in tnis case
{Yj, j € Z} exhibits long-range dependence.
The divergence of £°t_oo r(j) affects the behavior of the spectral density
hW = h Ej^-oo e~iXjr(j) as |A| —► 0: h(X) diverges at the origin when
1/2 <h\ 1 (see (7.2.27)).
• Suppose 0 < H < 1/2. Then ££L-oo |r(j)| < oo and E^=-oo r0') = °-
This last relation is due to the telescoping nature of r{j) (see (7.2.21)) and

336 SELF-SIMILAR PROCESSES 7.2
is consistent with the fact that the spectral density h(X) tends to zero as
|A| —» 0 (see (7.2.27)). Although there is no long-range dependence, the
case 0 < H < 1/2 is a singular one. Because the coefficient H(2H - 1)
is negative, the r(j)s are negative for all large j, a behavior sometimes
referred to as "negative dependence." It is easy to show that r(j) < 0 for
all j ^ 0, and thus
oo oo oo
0 = £ r{j) = r(0) + 2]Tr(j) = a2 + 2 JT r(j),
j=-oo j=\ j=l
■.e,Er=,r0>-^/2.
• Fractional Gaussian noise {V}, j — ...,—1,0,1,...} with H ^ 1/2
provides a counterexample to the usual central limit theorem. Indeed, for
T~ 13 7=i Xi t0 converge in distribution as TV —> co to a non-trivial limit,
one cannot choose d^ ~ vN as N —» oo, but one must choose djv ~ iV^
as N - co. In fact, ^EjL.^j = jhrBH(N) = BH{1). More
generally, the finite-dimensional distributions of ^ Xlj=i ^jj 0 < i < 1,
converge to those of Bh (t), 0<i<l,asAT-*oo. (Here [ ] means
"integer part" and £°=I ^ = 0.)
• Because of long-range dependence, normalized sums of polynomials of the
YjS may not converge to a Gaussian limit when 1/2 < H < 1 (Taqqu
1981).
The autocovariance r(j) of fractional Gaussian noise is given by (7.2.21) for
all lags j. In many modeling situations, however, one may want greater flexibility
in the choice of the r(j)s at small lags j while retaining the asymptotic behavior
r(j) ~ Cj2H~2 as j —► oo, which is characteristic of long-range dependence
when 1/2 < H < 1. This may be done in various ways. The simplest is to define
the moving average
j
Zj = 2_^ aj-k£k,
k——oo
where the e^s are i.i.d. N(0,1) and
r(j) = EZ0Zj = ]P a-kdj-k ~ Cj
2H-2
k——oo
as j —* oo. As the following theorem indicates, the ZjS satisfy the same type of
non-central limit theorem as fractional Gaussian noise.

7.2
FRACTIONAL BROWNIAN MOTION
337
Theorem 7.2.11 Fix 0 < H < 1 and let {Zj, j = ...,-1,0,1,...} be a
stationary Gaussian sequence with mean zero and autocovariancefunction r(j) =
EZqZj satisfying:
(i) Case 1/2 < H < 1:
r(i) ~ cj2H~2 as j —> oo >Wf/i c > 0;
(//) CaseH = 1/2:
OQ CO
]T|r(j)| <oo, ^ r(j) = c>0;
j=l j=-oo
fn'O Ca$e0< iJ < 1/2;
oo
r(j) ~ cj2H~2 as j —* oo wi7/2 c < 0 and Y^ r(j) = 0.
j=—oo
77ien the finite-dimensional distributions of {N~H ]T)j=i Zj, 0 < i < 1} co"~
verge to f«cse of{aoBH(t), 0 < t < 1} w/iere {Bh(*), 0 < t < 1} « standard
fractional Brownian motion and where
H~x{2H-\)-xc if \/2<H < 1,
c (/ if = 1/2,
-iJ-1(2H-l)-'c ifO<H<l/2.
PROOF: Since the ZjS are mean zero Gaussian random variables, it is sufficient
to show that for any constants 9U... ,8d, d> 1, andti,...,£<* G [0,1],
, d [Ntu] d d . [JVt„][iVt„]
u=l j=l u=l u=l j=l fc=l
converges as AT -> oo to £(En=i ^o-oBkC^))2- Hence, it is enough to show
that for any 0 < s < t < 1,
|Ns| [Nt] ,
j=i fc=i
Since, moreover, r(j) = r(—j), we have
[Ns][Nt] [Ns\[Na] [Nt][Ni]
EE-w-*) = HEE^-fc)+EE^-fc)
i=i it=i j=i k=i i=i fc=i
[JVt]-[Ns] [JVt]-[Ws]
- E E *■(*-*)}
J=l fc=l

338 SELF-SIMILAR PROCESSES 7.2
and [Nt] — [Ns] ~ N(t — s) as N —» oo, it is enough to prove
N N
^00N2Hllflr^-k) = al
Now
N N N-l j
A» = E E r0" - fc) = r(°) + E M°) + 2 £ r(i)}.
j=i fc=i i=i i=i
When 1/2 < H < 1, £i=i r(i) ~ (2# - l)-'^2""1 as j -> oo and thus, as
N —» oo,
AT
.4* ~ 2(2H - l)-lcJ2j2H~l ~ 2(2H)-l(2H - \)-xcNw
i=\
(Feller (1971), p. 281).
When H = 1/2, r(0) + 2^, r(j) = E^-oo r0') = ^ > 0 and we have
An ~ A^Uq as N —» oo.
The case 0 < H < 1/2 is more delicate. Since ^°1_00 r(j) = 0, we have
j oo
r(0)+2j>(j) = -2 53 r(j)~-2(2i/-l)-1ci2ff-1
i=l i=j+l
(Feller (1971), p. 281), and hence as N -> oo,
i4w ~ -2(2H- l)-'c^j2H-' ~ -2(2/f)-'(2if- l)-'cAr2i/. 1
i=i
There is another way to formulate the result of Theorem 7.2.11. Consider the
stationary sequence Z = {Zj, j = ..., — 1,0,1,...} of that theorem and define,
for each N > 1, the transformation
TN: Z -f TNZ = {{TNZ)U t = ...,—1,0,1,-...},
where
j (i+l)N
{TNZ)i^-^ ]P Zj, i = ...,-1,0,1,....
j=iN+\
Tm transforms the original sequence Z into a new sequence T^Z obtained by
summing the components of Z over successive blocks of size N and renormalizing
by
NH = (size of the block)".

7.2
FRACTIONAL BROWNIAN MOTION
339
The sequence of transformations {Tn}n>\ forms a semi-group because T^Tm =
Tnm- It is called the renormalization group with index H. The following corollary
is a consequence of Theorem 7.2.11.
Corollary 7.2.12 Let Z = {Zj, j = ..., —1,0,1,...} and o-0 be as in Theorem
7.2.11 and let Y = {Yj, j = ..., — 1,0,1,...} be standard fractional Gaussian
noise. Then, as N —> oo,
TNZ —► a0Y;
i.e., the finite-dimensional distributions of{{T^Z)j, j = ..., —1,0,1,...}
converge to those of{a0Yj, j = ..., -1,0, 1,...}.
Any sequence Y — {Yj, j = ...,—1,0,1,...} satisfying
TNY = Y for all N > 1 (7.2.28)
is called a fixed point of the renormalization group. Relation (7.2.28) is sometimes
used as definition of i?-self-similarity for stationary sequences (see, e.g., Major
(1981)).
Corollary 7.2.13 Fractional Gaussian noise is the only Gaussian fixed point of
the renormalization group.
Proof: That fractional Gaussian noise is a fixed point follows from Corollary
7.2.12. This fact can also be readily verified by noting that for any 0\,...,dd, d>
l.andiV > 1,
d d {i+\)N
i=l i=\ j'=iAf+l
d 1
d
d
- E<w-
i=l
Fractional Gaussian noise is the unique fixed point because fractional Brownian
motion is the unique Gaussian i?-sssi process. 1
Although fractional Gaussian noise is not the only Gaussian sequence with
long-range dependence, it is the only one which is a Gaussian fixed point of the
renormalization group, and as such, it serves as an important paradigm.

340 SELF-SIMILAR PROCESSES 7.3
7.3 General characteristics of processes that are
a-stable and H-sssi
We now turn to a-stable #-sssi processes {X(t), ( el} with 0 < a < 2.
The admissible values of H are H £ (0,1/a] if 0 < a < 1 and H € (0,1] if
1 < a < 2 (Corollary 7.1.11). We will see in the next sections that, in contrast to
the Gaussian case a = 2, the property of "i7-sssi" does not typically determine the
finite-dimensional distributions. It imposes, however, a number of restrictions.
Lemma 7.3.1 Let {X(t), t e K} be a-stable, 0 < a < 2, and H-ss, and let T
denote the spectral measure ofthe random vector (X(t\),... ,X{td))inM.d. Then,
for any a > 0, the spectral measure ofthe random vector (X(ati),..., X(atd))
is aaIiT.
Proof: Use Theorem 2.3.1. I
Lemma 7.3.2 Let {X(t), t e R} be a-stable with 0 < a < 2 and H-ss. Then
X(0) = 0 and for fixed t € K. X(t) ~ Sa(ax(t), 0X[t), px(t)) where
t i M
^X(t) = 1*1 °X(1),
Px{t) = (sign«)/3x(i),
r |i|"(signt)Mx(1) ifa^l,
Mx(t) = {
{ |i|*(sign t)(px{l) - l|t|»(ln |t|ffW(1)i9x(0) if a = 1.
Proof: Use (1.1.6) and the relation X(t) = |t|HX(signt), valid for fixed i. I
Proposition 7.3.3 Let X — {X(t), t 6 R} be an a-stable H-sssi process with
0 < a < 2.
(0 IfH^ 1, then X is strictly stable.
(ii) IfH — \anda ^ 1, then there is a constant p such that {X (t) —pt, t € M}
is strictly stable.
PROOF: Set at = o-X(t), Pt = Px(t) and H = MX(t) and recall that the strict
stability of the finite-dimensional distributions is equivalent to the strict stability
of all the marginals (Corollary 2.4.2). We shall consider separately the three cases
(i)H jt 1, a/ l,(ii)#V la = L and (hi) H = 1, a# 1.
(i) Case H =fc 1, a ^ 1. Since a ^ 1, we need to show that ^it = 0 for all
t. The stationarity of the increments implies X(2) — X(l) = X(l) and hence by

7.3 GENERAL CHARACTERISTICS OF q-STABLE tf-SSSI PROCESSES 341
Example 2.3.4, /i2 — Mi = Mi> '-e-> M2 — 2fi\. On the other hand, H-ss implies
/z2 = 2Hn\ by Lemma 7.3.2. Equating the last two relations yields pL\ — 0 and
hence fxt = 0 for all i € R by Lemma 7.3.2.
(ii) Case H ^ 1, a = 1. Since a = 1, it is enough to show that
Pt = 0 for all i. We firstly show that 0X = 0. Let T denote the spectral
measure of the vector (X(2),X{1)). The relation X(2) - X(l) = X(l) implies
crx(2)-x(i)/3A'(2)-x(i) = ft A • Since, on the other hand (see Example 2.3.4),
cx(2)-x(i)/?x(2)-x(i) = / |s2-s1|sign(s2-s1)r(ds)
-/s2
= /"(*2-*i)r(ds)
•/s2
= ct2/?2 - a2/3i,
we have cr2/32 = 2<j\(3\. By #-ss, ct2A = 2Ha\(5\ and hence 2<7i/3i = 2Hcr\(5\,
which is possible only if/3| = 0, and thus /3t = 0 for all i by Lemma 7.3.2.
(iii) Case H = 1, a ^ 1. Then {X(i) - pt, t 6 R} is strictly stable
(Theorem 2.4.1) and /j.t = ty,\, ieM (Lemma 7.3.2). I
One can make no general statement in the case H = 1 and a = 1. For example,
ifX(l) ~X1(cr1,A,Mi) with/3, ^ 0, then the process {X(t) =tX(l), (£»},
is 1-stable, H-sssi but is not strictly stable, even if ^1 = 0, because fit = P\ ¥" 0-
Corollary 7.3.4 Let {X(t), t S M} be a-stable, 0 < a < 2, and H-sssi. Fix
U,t2, t £i. Then
(')ifa^l:
<TX{ti)-X(t,) = 1*2-tl| CTX(l),
Px(t) = (signt)/?x(i).
VX(t)
0 ifH^l,
tf*x(i) ifH = l;
(ii)ifa- \andH ^ 1:
<?x(t2)-x(t{) = \h-h\ <7x<i)>
Px{t) = 0,
Mx(t) = |<|H(signt)Mx(i).

342
SELF-SIMILAR PROCESSES
7.3
that is,
1 2 /"
Hx(t) = -Mtf(signt)2_2H- / (S2 -si)ln|s2 -si|rx(2)iX(1)(ds).
Proof: The only new statement concerns Mx(t) when a = 1 and H £ \,
which we now verify. Since Asqi) = 0 (Proposition 7.3.3), we have ^x(t) =
|i|H(sign t)nx(\) by Lemma 7.3.2. To devise an expression for fJ-x(i), use
Mx(2)-x(i) = Mx(2) - Mx(i) - ~ / («2 -si)ln|s2-S||rx(2),x(i)(ds),
Relation (2.3.5), Hx(2)-X(\) - Mx(i) and Mx(2) = 2H//x(i). Combining these
expressions yields the desired result. I
Wheni/ ^ l,theparameter^x(t)dependsonaandif,andalsoonrx(2),x(i)
if a — 1. What is important, however, is the following implication:
Corollary 7.3.5 If{X(t), t € R} is a-stable H-sssi with if ^ 1, then {X(t) +
/x?, t € T} cannot be H-sssi unless the "shift" /i° is identically equal to zero.
if-sssi processes are typically defined by their integral representation, a more
"physical" quantity than the joint characteristic function. Suppose, then, that the
a-stable if-sssi process {X(t), tef} has the integral representation
*(t)= [ ft(x)M(dx), te:
JE
where M is an a-stable measure with skewness intensity 0(x), x £ E, and
control measure m and JE \ft{x)\am(dx) < oo for all t € R. 7 What restrictions
on ft, i6l, does the condition of "H-sssi" impose?
Proposition 7.3.6 Suppose /?(•) == 0 if a = 1. Then the a-stable process
{X(t), t £ R} « H-sssi if and only if for any integer d > 1 and real
numbers Oj,tj, j = 1,..., d, the integrals
r d
a~aH $>(/ati+fc(s)-A(aO)
J B ■ 1
'B i=l
x(d:r)
'When a = I, we must also suppose
|/(x)/?(i)ln|/(x)||m(dx)<oo,
/.
but this will not be necessary because, in the sequel, we will always set /3(-) = 0 when a = 1.

7.4
LINEAR FRACTIONAL STABLE MOTION
343
and
a-aH / \j2ej(fatj+h(x) - fh(x))] " P(x)m(dx)
are independent ofa>0 and h € R.
PROOF: H-ssrequires Y?j=\ QjX(atj) = aH £^=, 0jX(fj)andthestationarity
of the increments requires the distribution of J2j=\ ®i (-^(*i + h) - X(h)) to be
independent of h. Now use the expression for the characteristic function given in
(3.2.2). I
Remarks
1. Different representations may define the same process and, hence, in the
sequel it will be important to know when this happens.
2. In the case a = 1, we choose the skewness intensity /3(-) = 0. Then the
process X is, in fact, Si5.
7.4 Linear fractional stable motion
There are many different extensions of fractional Brownian motion to the a-stable
case. The one which is most commonly used is the linear fractional stable motion
(also called linear fractional Levy motion). It is defined as follows.
Definition 7.4.1 The linear fractional stable motion (LFSM) is the stochastic
process {LajH(a, b;t), — oo < t < oo} given by
/oo
fa,H{a,b;t,x)M(dx), (7.4.1)
-OO
where
/Q,„(a,M,x) = a(((t-x)+)H-1/Q-((-x)+)^'/a)
+6({(t-*)-)ir-I/a-((-x)_)H-,/a), (7.4.2)
and where a, b are real constants, |a| + |6| > 0,0 < a < 2, 0 < H < \,H ± 1/a,
and M is an a-stable random measure on M with Lebesgue control measure and
skewness intensity /3(x), —oo < x < oo, satisfying:
(i)/?(-)=0ifa = 1,

344
SELF-SIMILAR PROCESSES
7.4
(ii) for all integers d > 1 and real 9j, tj, j — 1,..., d,
I (^2Qj{f«,H{aMtj,x) - fa,H(a,b;0,x)))<a>'p(cx + h)dx (7.4.3)
is independent of c > 0 and —oo < h < oo.
The representation of LFSM is based on the representation (7.2.7) of fractional
Brownian motion. Here, the exponent is H — l/a instead of H — 1/2. Therefore
LFSM reduces to fractional Brownian motion if one sets a = 2. Figures 7.2
and 7.3 show the kernel fQ,H(a,b;t,x) for a = 1, b = 0 and various values
of t and d = H — 1/q; and Figure 7.10 shows it for a = b = 1, t = 1 and
d = H- 1/q = ±0.05.
Remarks
1. The process {LQtH(a,b;t), —oo < t < oo} is well denned because
fToo \fa,H(a, b; t,x)\adx < oo, as in Example 3.6.5.
2. La,H(a, b; t) is not defined for H = 1 because J^ |/Q,i(a; b; t, x)\adx =
oo, t 7^ 0. Hence 0 < H < 1 is a necessary condition for LQih to be well
defined.
3. The well-balanced linear fractional stable motion
La,H(h Ut) = f (\t - x\H~x'a - \x\"-"a)M{dx) (7.4.4)
introduced in Example 3.6.5 corresponds to the choice a = b = 1.
4. The SetS case corresponds to /?(•) = 0.
5. We suppose /?(•) = 0 when a = 1 in order to provide a simple condition to
ensure that the process is strictly stable.
6. Condition (7.4.3) is, in general, very difficult to verify. In practice, we take
P constant, typically /3(-) = 1. The condition then holds trivially with such
a choice of /3.
Proposition 7.4.2 The LFSM is H-sssi.

7.4
LINEAR FRACTIONAL STABLE MOTION
345
PROOF: It is sufficient to verify the conditions of Proposition 7.3.6. For example,
for any c > 0 and -co < h < oo,
c~aH (y^OjUa.Hfa'hctj +h,x) - fa,H(a,b;h,x))J /3(x)dx
r(d \<a>
= c aH ^29j(faiH(a,b;ctj,cx) - fa,H(a,b;0,cx)) 0(cx + h)dcx
J-oo ^=1 J
Y^ejUatf{aMtj,x) - fa,H(a,b;0,x)) P(cx + h)dx
which does not depend on c, by Assumption (7.4.3) if a ^= 1 and by the choice
f3 = 0 when a =1. I
Remark. By analogy to the case a = 2, we say that the increments of LFSM
have long-range dependence when H > 1/a and negative dependence when
H < 1/a. There is no long-range dependence when 0 < a < 1 because H is
constrained to lie in the interval (0,1). The value H = 1/a lies on the boundary
between long-range and negative dependence. Processes with H — 1/a will be
considered in Sections 7.5 and 7.6.
Proposition 7.4.3 // {X(t), -oo < t < oo} is LFSM, then for each fixed t €
R, X(t) has a SQ(crt, /3t, 0) distribution, where
at = Ka,H(a,b)\t\H
and
KtH(a,b) = (|a|Q + \b\a) fX ((1 +x)H-^a - xH-l'a)adx
Jo
+ f \a{\-x)H-'la -bxH~l>a\adx. (7.4.5)
Jo
If, moreover, the skewness intensity /?(•) equals a constant (3, then
&=vJt *><a> - 6<q>) r K1+*)""i/a - xH-i/a)afc
+ f (a(l - x)"-"a - bxH-x'a)<a>dx.
Jo

346
SELF-SIMILAR PROCESSES
7.4
PROOF: We establish the expression for j3t. (The expression for at is obtained
in a similar fashion.) Assume, without loss of generality, that t > 0. If 0(x) = /?,
then, by Property 3.2.2,
1 f°°
0t = — / (fQ,H(a,b;t,x))<a>p(x)dx
at J-oo
= 4 r (fa,H(^b;t,x))<a>dx.
at J-oo
Now use the expression (7.4.2) for /Q,i/(a, 6; t, x). If (3 ^ 0,
ff«"£ = o<a>/"° {{t~x)H-"a-{-x)H-x'a)adx
+ f (a(t - x)H-"a - b xH-x'a)<a>dx
Jo
({x-t)H-"°-xH-l/a)<a>dx.
The third integral can be written as
/oo poo
(x"-l/a -{x- t)H-"a)adx = - / ((* + u)H-"a - uH-x'a)du,
so that
a. . rrC. ... .... ^ .
ax
a^ = |i|a"{(a<a> - 6<a>) f° ((1 + x)H-x'a - xH-x'a)ac
+ fl (a(l - x)H-'/Q - b xH-"a)<a> dx). I
The process
\X{t)-— —TrLaiH(a,b,i), -oo<t<ool,
where the constant Katjj(a,b) is given in (7.4.5), is called a standard linear
fractional stable motion (standard LFSM). It is normalized in the sense that the
scaling parameter of X (1) equals 1.
We want to show now that the (standard) linear fractional stable motions
{La,H(a-,b;t), t € K} are typically different for different a and b. Theproofuses
the following result (Lemma 4 of Kanter (1973)).

7.4
LINEAR FRACTIONAL STABLE MOTION
347
Lemma 7.4.4 (Kanter). Let f,g € La(E). If for some 0 < a < 2, any integer
d, and any real numbers 6\,... ,6d, t\,... ,td,
/oo d i-oo d.
| J2Ojffa + x)\adx - / | £ejg(tj + x)\adx,
-oo J=, J-oo j=l
then there exists a number e equal to — 1 or +1 and a real number r such that
f(x) = eg(x + t) a.e. x.
Theorem 7.4.5 Let 0 < a < 2, 0 < H < 1, H j= 1/a, and let a, a', b, b' be real
numbers satisfying \a\ + \b\ > 0 and\a'\ + \b'\ > 0. Then the standardLFSMs
La,H{o,,b;t) , —oo < t < oo,
KaM{a,b)
and
-r-r^LatH{o.'.b';t) , -oo < t < oo,
KaiH(a',b'Y
have identical finite-dimensional distributions if and only if one of the following
conditions holds:
(i) a = a' = 0,
(ii) b = b' = 0,
(Hi) a, a\ b', b' are non-zero and a/b — a'/b'.
Proof: Suppose that one of the three conditions holds, for example, (iii). Then
1 ,,..*<* b
-LaiH{a,b;t) = — t—-LaM(-r, l;t)
I Ka,H{a,b) \b )
has same finite-dimensional distributions as
K,
^^''•^-idbMi.'")
because they differ by a multiplicative constant which must be equal to 1 since
both processes have been normalized.
To prove the converse, suppose
K-*H{a,b)La,n{a,bt) ^d K^H{a',b')La,H{a!,b';t)
have the same finite-dimensional distributions. Then the stationary processes
Y{a,b,t) = K-*H(a,b)[LaM(a,b;t + h) - La>H(a,b;t)}, -oo < t < oo,

348
SELF-SIMILAR PROCESSES
7.4
and
Y(a',b', t) = K~xjj{a', b')[LatH(a',b'; t + h)- Laj{.a',b'; t)], -co < t < oo,
also have identical finite-dimensional distributions, for any given h > 0. We want
to show that this implies that one of the conditions (i)-(iii) holds. The process
Y(a, b; t) can be represented as
/oo
g(a,b;t-x)M(dx),
-CO
where
g(a,b;t) = a[(t + h)^~1/a - t^1'"} + b[(t + hf_-l'a - tH_-x/a\.
Since Y(a, b; t) and Y(a\ b'; t) have the same finite-dimensional distributions,
Eexp^iJ20jy(a,b-,tj)j\=\EtxpUYJOjY(a',b';tj)\
i=i
i=i
i.e.,
/oo d
EeiK*!H(a>b)9(a,ktj -x) dx\
■°° j=\
/oo d
EW,»(«'. b')9W\ b'; tj - xtfdx),
■°° i=i
for all integers d and real numbers 6U... ,9d,ti,... ,td. Hence, by Kanter's
Lemma 7.4.4, there exists a real number r and ane e {+1,-1} depending on
h, a, b, a', b' such that
K,h(^ %(a> b> *) = e#a,'tf(a'> b')9(a', b'; t - t) a.e. t,
K-!„(a,b) [a[(t + h)%-,/a - t^~1/a} + b[(t + hf_-"a - tH_-x/a)
= <*«,>'.*')[Ait + h - rf+~X'a -(t- r^-]/a}
i.e.,
+ &'[(* +A-r)?-|/a-(t-r)?-1/a]
a.e. t.
(7.4.6)
Since both sides of (7.4.6) represent functions continuous everywhere except at
the four points t = — h, 0, r — h, r, Relation (7.4.6) can be extended by continuity
to all t not equal to these four points.

7.5
Q-STABLE LEVY MOTION
349
We claim that this implies
Ka,H[a,b) eKa}H(a',b>)' KQ,n(a,b) €Ka,H{a',Vy (7A7)
Indeed, differentiating (7.4.6) with respect to t ^ —h, 0, r - h, t, we obtain
K!n(*>b) [a[(t + h)H+-l/a-' - t^-l/a-1} - b[(t + hf_-x/a-x - tH_-l/a-1)
= eK~;H(a', b') [a'[(t + h - r)^-1. - (t - t)^1^}
-6,[(t + ft-T)?"1/a-I-(i-T)?-,/a-1]] (7.4.8)
for all t except t = —h, 0, t — h, r. Suppose, without loss of generality, that
a =£ 0 and let t j 0. Then the left-hand side of (7.4.8) tends to ±oo. For the
right-hand side to do the same it is necessary that r = 0 or r = h. To see that
r 7^ h, set r = h and let t [ h. This makes the left-hand side of (7.4.8) tend to a
finite number and the right-hand side tend to ±oo, which is not possible. Hence
t = 0 and (7.4.7) holds. Relation (7.4.7) implies that one of the following
(i) a = a' = 0
(ii) b = b' = 0
(iii) a, a', b, b' are non-zero and
Kg,H{a,b) _ a _ b _
'KatH(a',b') a' V
7.5 a-stable Levy motion
The linear fractional stable motion (LFSM) was defined for H ^ \ja. Its
definition can be extended to the case H = I/a in two different ways. One leads
to the a-stable Levy motion (discussed here), the other to the log-fractional stable
motion (discussed in the following section).
Definition 7.5.1 An a-stable Levy motion, 0 < a < 2, is a process {X(t), t £ R}
with stationary independent increments having a strictly a-stable distribution.
By independent increments, we mean, as usual, that the random variables X(tj) —
^(ij-i), j = 2,..., d, are mutually independent for any -co < ti < <2 < ■ • • <
td < oo, d > 3.

350 SELF-SIMILAR PROCESSES 7.5
Theorem 7.5.2 Let 0 < a < 2 and let M be an a-stable random measure on
R with Lebesgue control measure and constant skewness intensity /3(-). (Assume
/?(•)= 0i/a= I.) Then
X(t)
J*M(dx) t>0,
/°(M(dx) t < 0,
is a-stable Levy motion.
M*)
0 t
Figure 7.8: Integration kernel for
Brownian motion (a = 2) and a-stable
L6vy motion (0 < a < 2) when t > 0.
(7.5.1)
Remarks
1. The choice /?(•) constant and /?(•) = 0 if a = 1 is made for practical
reasons. X, with /3(-) = 0, is the SaS Levy motion introduced in Example
7.1.3.
2. The definition extends to a — 2. The 2-stable Levy motion is obviously
Brownian motion.
3. For t > 0, Brownian motion (a = 2) and a-stable L£vy motion (0 < a <
2) can be represented as J^ l[0it](i)M(dx) = M([0, t]) (see Figure 7.8).
The proofs of the previous theorem and of the following proposition are
straightforward.
Proposition 7.5.3 The a-stable Levy motion is H-sssi with
1 t\
H
a£(r°°)-

7.5
a-STABLE LEVY MOTION
351
By allowing the value H = 1/q and using the convention 07 = 0 for all
values of 7, Definition 7.4.1 of LFSM can be extended to include the a-stable
motion. Indeed, for t > 0,
fa,i/a(a,b;t,x)
= a(l(-oo,t)(z) - l(-oo,o)0)) + b(l(t<oo)(x) - l(o,oo)(a;))
= al[o,t)(a:)-61(0,4](:c).
Integrating this kernel with respect to the random measure M, we obtain
(a — 6) JQ M(dx). This differs from (7.5.1) only by the multiplicative factor
a — b.
Setting, in the same way, H — 1/2 in the representation (7.2.4) of fractional
Brownian motion yields Brownian motion.
Observe that although the a-stable Levy motion is ff-sssi with H = 1/a €
(1/2,00), the self-similarity parameter H of the LFSM restricted to the interval
(0,1). Observe also that Theorem 7.4.5 is not valid for a-stable LeVy motion: the
parameters a and b of that process yield here only a multiplicative factor a — b.
Are there other a-stable (l/a)-sssi processes, besides a-stable L6\y motion?
The following theorem shows that the answer is in the negative when 0 < a < 1.
Theorem 7.5.4 The only non-degenerate a-stable 1 /a-sssi processes with 0 <
a < 1 are the a-stable Levy motions.
Proof: Let {X(t), t € R} be a non-degenerate (i.e., A'(l) ^ 0 a.s.) a-stable
1/a-sssi process with 0 < a < 1. Corollary 7.3.4 implies that {X(t), t S
1R} is strictly a-stable. Let at denote the scaling parameter of the a-stable
random variable X(t). Fix arbitrary s\ < sz < t\ < £2- The random variables
X(si), X(s2), X(t\) and X(t2) are jointly strictly a-stable, and thus, by the
representation theorem 3.5.6, there are functions /s,, /S2, /t, and ft2 in La([0,1])
such that
WsO.XteJ.XM.Xfo))^ (J fj(x)M(dx), j = sl,a2,tut2),
where M is an independently scattered a-stable measure on ([0,1], B) with
Lebesgue control measure and skewness intensity /?(•) = 1. Then
(t2 - s,k = <-Si = [ i/tl(x) - /s,(*)r^
Jo
< I \fS2(x)-U(x)\adx+ I\ft>{x)-fSl{x)\°dx+ [ |/t2(x)-/t,(x)r<*r
./o Jo Jo
(7.5.2)

352 SELF-SIMILAR PROCESSES 7.6
= <_„ + <_„ + <rg_t, = (s2 - a,K + (i, - s2)< + (t2 - iOaf
Here we have used the self-similarity and the stationarity of the increments
of {X(t), t € M} (Corollary 7.3.4). Thus, the inequality in (7.5.2) is actually an
equality, implying
(f.>(x) ~ /., (x))(ftl(x) - ft> (x)) = 0 a.e.
It follows from Theorem 3.5.3 that X(s2) - X{si) and X(t2) - X(ti) are
independent for any si < s2 < t\ < t2 and, since for jointly stable random variables
pairwise independence is equivalent to total independence (Corollary 3.5.7), we
conclude that {X(t), t € R} has independent increments. Thus {X(t), t € R}
is an a-stable L6vy motion. I
When 1 < a < 2, the a-stable L6vy motion is not the only 1/a-sssi process.
If a = 1, the process X(t) = tX, where X is a 1-stable random variable, is 1-sssi.
If 1 < a < 2, there is the log-fractional stable motion defined in the following
section and also the processes defined in Relation (7.14.1) (see Exercise 7.12).
7.6 Log-fractional stable motion
We indicated at the beginning of Section 7.5 that there are several ways to extend
Definition 7.4.1 of LFSM to the case H = I/a. One way yields the a-stable L6vy
motion. Here is a different extension: set a > 1 and a = b = 1 and consider
1 r°° (\t-x\H-^Q - 1 \x\H~x/a - 11
5^ wo. >.«) - If *'_,/a - ^biT^}"^
The fact that for any real u ^ 0,
|u|H->/«_i e("-i/*)inM_i
lim LJ5—— = lim —— = In \u\
H-*\/a H~\/a if—i/q H-\/a
suggests the following definition:
Definition 7.6.1 Set 1 < a < 2 and let M be defined as in Definition 7.5.1 (with
/?(•) = P constant). The process
AQ,i/Q(t) = / (ln|t-x| -ln\x\)M{dx), -co <t < oo, (7.6.1)
J—CO ^ '
is called log-fractional stable motion (log-FSM)J

7.6
LOG-FRACTIONAL STABLE MOTION
353
In 1* -x\ — ln|x|
6
Figure 7.9: Kernel of the log-fractional stable motion (see (7.6.1)) plotted for several
values of t.
The kernel is illustrated in Figure 7.9 and it is compared in Figure 7.10 to that
of the well-balanced fractional stable motion. To verify that the process AQil/Q(£)
is well defined, we will check that its integrand belongs to La. The function lnQ \x\
is integrable around x = 0, and, as x —> oo, the integrand behaves like (l/x)Q,
which is an integrable function for a > 1. The process {AQ]i/Q(i), t S M} is
therefore well defined.
Remark. The log-fractional stable motion is not defined for a < 1 because x~a
is not integrable as x —* oo.
Proposition 7.6.2 The log-fractional stable motion is H-sssi with H = I/a.
Proof: Since H = 1/a and /3 is constant, the conditions of Proposition 7.3.6
are satisfied. For example for any c > 0 and -oo < h < oo,
c~a" I E ^'(ta \cti +h-x\-\n\h- x\j\ <a>/3dx
oo - d
= / |E 6j(In \ctj - cx\ - In |ce|)
1 <a>
/3dx
j = l
oo - d
= / E ^(ln I*j _ xl ~ln I1')] <Q>/3da
■'-o0j=l
It is sometimes called log-fractional Levy motion.

354 SELF-SIMILAR PROCESSES 7.6
Figure 7.10: Comparison, at t = 1, of the kernel In \t — x| - In |x| of the
log-fractional stable motion and the kernels |t — x|d — |x|d, d = H — 1/q, of the
well-balanced fractional stable motion for small values of d.

7.7 THE REAL HARMON1ZABLE FRACTIONAL STABLE MOTION 355
is independent of c and h. I
We noted that the definition of the log-fractional stable motion cannot be
extended to a < 1. It can be extended, however, to a — 2, but in that case, one
does not obtain anything new. Since the resulting process is Gaussian and ff-sssi
with H = 1/2, it must be Brownian motion.
When 1 < q < 2, the log-fractional stable motion does not have
independent increments and therefore is not the a-stable Levy motion. To see
that the increments are dependent, set AQ)1/Q(1) = f^°oof(x)M(dx) and
AQ,i/a(2) — AQil/Q(l) = J^ g(x)M(dx) and observe that one does not have
fg - 0 a.e. (Theorem 3.5.3).
Finally, note that Definition 7.6.1 of log-fractional stable motion involves a
"well-balanced" representation analogous to that of the linear fractional stable
motion (7.4.2) with a = b = 1. A representation that is not "well-balanced"
would not yield a self-similar process. Indeed, set Ino x equal In x if x > 0 and 0
if x < 0. Let a and 6 be two real numbers. Then the process
/ (a[\n0(t - x)+ - lno(-x)+] + &[ln0(i - x)_ - ln0(-x)_] W(dx)
J—00
is not self-similar unless a = b, in which case we have aAQ)i/Q(i). For example,
if a = 1 and 6 = 0, the scale parameter crt of the process at a fixed t > 0 satisfies
a? = / [ln(t + x)-\nx}adx+ / \\nx\adx
Jo Jo
J/*oo A
' [ln(l +x)-\nx}adx + t / |lnx + lnt|ada:
0 Jo
* tof.
7.7 The real harmonizable fractional stable motion
We saw in Proposition 7.2.8 that fractional Brownian motion has the
"harmonizable representation" J^ ^l\x\<H-xI^M{dx) where M = Af<'> +»M<2>
is a Gaussian complex measure whose real and imaginary parts M^ and M^
are independent with Lebesgue control measure.
The corresponding representation in the case a < 2 involves a random measure
M whose real and imaginary parts M^ and M^ are dependent. The random
measure M, is, however, isotropic, i.e., its circular control measure k on R x 52
equals 7717, where 7 is the uniform probability measure on 82- (Refer to Section
6.3 for details.) Here, m is the Lebesgue measure on R.

356 SELF-SIMILAR PROCESSES 7.7
Definition 7.7.1 The real harmonizable fractional stable motion (RHFSM) is the
process
{/■oo ixt _ i ,
Ha,H(t) = Re / — \x\-H+l-VaM(dx), teR , (7.7.1)
J— oo *■£ '
where 0 < a < 2, 0 < # < 1 and where M is a complex isotropic SctS random
measure with Lebesgue control measure.
We must first verify that thejntegral in (7.7.1) is well defined. Recall that an
integral /(/) = Re /^ f(x)M{dx), where f = fW + z/(2) (/,, /2 real-valued)
and where M is isotropic (with control measure m) has characteristic function
£expz0/(/)=exp{-|0|aco [ \f(x)\am(dx)\ (7.7.2)
where Co = ^ fQ*(cos4>)ad<f> (Corollary 6.3.2). Applying this to Ha,H(t), we
obtain its finite-dimensional characteristic function
d
£exp{i £ fljWa,//(*,-)}
= exp{-co J_jJ^y—Zlx-^-^dx}
= exp{-co f \J29i(eiXtj ~ \)\ax~aH~Xdx\. (7.7.3)
J-oo j=l
As |i| —> oo, the integrand behaves like \x\~aH~i, which is integrable at infinity
because H > 0; as x —> 0, the integrand behaves like \x\a~aH~l, which is
integrable at zero because H < 1. Therefore the process {7-£q,h (£), —oo < t <
oo} is well defined.
Remarks
1. Writing
/oo
f{x)M{dx)
-OO
= P /(l)[»M(l)(dx) - f /<2)(x)M(2)(dx),
J-oo Jb
we obtain the following alternative representation for the process Ti.atH-
naH(t) = r —M-w-^MVXdx)
J-oo X

7.7 THE REAL HARMONIZABLE FRACTIONAL STABLE MOTION 357
+
f
J — c
1 — COS xt,
-H+l-i/a
M{2){dx), (7.7.4)
where M'1' and M^ are the real and imaginary parts of the isotropic
random measure M. The factors of M^ and M^ were plotted in Figures
7.4 and 7.5 for t = 1 and d = H - \/a = ±0.2. The measures M(1> and
M® are dependent, but they become independent when a — 2 (Section
2.6).
2. Comparing Relation (7.7.4) with (7.2.16), we see that by setting a = 2, the
process Ha,H(t) becomes CB}j(t) where C is a constant and where the
fractional Brownian motion Bfj(t) is given by its harmonizable
representation (7.2.12). (See also Exercise 7.14.)
Proposition 7.7.2
The real harmonizable fractional stable motion {7ia)#(i), —oo < t < co}
is H-sssi.
Proof: Exercise 7.13. I
The following proposition shows that there is nothing to gain by considering
the process
/oo ixt _ 1 / N
. (Q(x+)-g+1-'/"+b(x_)-g+1-1/")M(&),
-oo lX V J
where o, 6 are real numbers.
Proposition 7.7.3 The finite-dimensional distributions of {riatH(a,b;t), —oo <
t < oo} are equal to those of
{(Wl±tt)"V„m, -„<«<4
Proof: Fix d > 1, 9\,..., 9d, U,..., td, and let Co be the constant introduced
in (7.7.2). Writing Eexp{iJ(/)} = exp{-||J(/)||2}, we have
I>Wa,«(a,MiC
j"=i
ix
/oo d
/oo <*
52^(e^-l)
■°° j=i
{x+)~H+x~XIa +b{x-)-H+x~x/a\adx
\x\-*{x+y
-aH+a-\
dx

358 SELF-SIMILAR PROCESSES 7.8
/oo d a
^^■(e^-1) |x|-Q(x_)-
ldx
since x+x_ = 0. Clearly, each of these last two integrals equals
5 SZo I E"=i ^(e"^ - l)\a\x\~aH-ldx and therefore, by (7.7.3),
We have been considering the real harmonizable fractional stable motion.
Would a linear combination of the real and imaginary part, namely
/oo txt 1
Z-^±\x\-H+i-l'aM(dx)
-oo ia;
/oo ixt 1
i_ Ixl-w-i-i/a^da;)
•oo lx
yield new i?-sssi processes? The answer is in the negative (Collorary 6.3.3).
Finally, in sharp contrast to the Gaussian case, we have the following
consequence of Theorem 6.7.2.
Theorem 7.7.4 The linear fractional stable motion defined in (7.4.1) and the real
harmonizable fractional stable motion defined in (7.7.1) are different processes.
Proof: If the two processes have the same finite-dimensional distributions (up
to a multiplicative factor), then their increments (which are stationary processes)
would also have the same finite-dimensional distrubutions. This would contradict
Theorem 6.7.2. I
7.8 Complex harmonizable fractional stable motion
Here, we consider the complex harmonizable fractional stable motion (CHFSM)
{C,„(a,6;t)
= f° ^!-ll(a(x+)-H+1-1/Q + K^-)"/f+1-1/Q)M(dx)) t€R},
(7.8.11)
where 0 < a < 2, 0 < H < 1, a > 0, 6 > 0, a + b > 0, and where M
is a complex isotropic random measure SaS with Lebesgue control measure as

7.8 COMPLEX HARMONIZABLE FRACTIONAL STABLE MOTION 359
in Definition 7.7.1. We studied in Section 7.7 the real harmonizable fractional
stable motion T~ia,H(t) = ^e CQ,#(1, 1;£), and we noted that Re Ca>f{(a,b;t),
or for that matter Im CQ)H(a, b; t), is again Ha,H{t) UP to a multiplicative
constant. In contrast, different choices of a and b typically yield different complex
harmonizable fractional stable motions. To show this, we need firstly to derive
the finite-dimensional characteristic function of {CQ,//(a, 6;f), —oo < t < oo}.
Fix t\,..., td and let Zj = 0\' +i$\\ j = 1,..., d be arbitrary complex
numbers. Applying Theorem 6.3.4, we obtain
d d
i=i
Eexv{i(j2e^ ReCQ,H(a,6;t,)) +]T><2) ImCQ,tf(a,Mj)}
j=i
d
= exp{-H E^c«."(a'6;t;)lla}' (7-8-2)
where
iiX>cQ,*(M;*i)ii;;
/oo d.
|$>i(e<Iti - l)|a(|a|°(i+)-Q-H-Idi+|6r(i_)-Qfl-,)da:
■°° j=i
/■oo ^ d
as in the proof of Proposition 7.7.3, with cq — j^ J0 ^(cos ()>)ad4>.
Proposition 7.8.1 The complex harmonizable fractional stable motion
{Ca,H{a,b;t), teR}
is H-sssi.
Proof: It is sufficient to show that for any real c\ and c2, {c\ Re Ca,H (a, b; t) +
c2 Im Co,,i/(a, b; t), t6l} is if-sssi. This, however, follows from (7.8.2). I
The normalization factor used to standardize the process CQ,//(a,6;-) is
||CQ,i/(a,6; 1)||Q. Setting d= 1, z\ = 1 andii = 1 in (7.8.3), we obtain
/»oo
\\Ca,H(a,b; 1)||S = co / (aa\eix - l\a + ba\e~ix - \\")x
Jo
aH-\
dx

360
SELF-SIMILAR PROCESSES
7.8
X
2 sin -
2
x-aH-xdx
= (aa + 6Q)co f
Jo
/•OO
= (aa + ba)co2a^-HU | sin x\ax~aH-xdx. (7.8.4)
./o
The standard CHFSM is the process
CaiH(a,b;t) _ CaM(a,b\t)
\\CaM(a,b;l)\\a (a<*+b<*y/<*2(i-")(c0f™\smx\*x-aH-idxy/a'
Note also that the parameters a and b in the definition (7.8.1) of the complex
harmonizable fractional stable motion were restricted to K+. This is because
choosing them in K would not have yielded new processes as Relation (7.8.3)
demonstrates.
We shall now show that each ray through the origin of the parameter space
(a, b) € R+ defines a distinct complex harmonizable fractional stable motion.
Theorem 7.8.2 Let 0 < a < 2, 0 < H < 1, and let a, a', 6, b' be non-negative
real numbers satisfying a + b > 0 and a' + b' > 0. Then the processes
and
{(^-py^CQ'*(o''6';f)' -°° < * < °°}'
have identical finite-dimensional distributions if and only if one of the following
conditions holds:
(i) a = a' = 0,
(ii) b = b' = 0,
(Hi) a, a', b, b' are non-zero and a/b — a'/&'.
Proof: The "if" part follows immediately from (7.8.3). Consider now the "only
if" part and, in particular, case (iii) (the others are the same). Setting, without loss
of generality, a = a' = 1, suppose
(1 +6°)-,/aCQlW(l,6;-) = (1 +6"T1/QCQ,H(l,b';-) (7.8.5)
for some 6 and b'. We want to show that this implies b = b'.
We will work not with CQ|h(i> b, •) but with
7(1,6;*)= J [CaM(l,b;t)-CatH(l,b';t + u)}eudu, -co < t < oo,
J — OO
(7.8.6)

7.8 COMPLEX HARMONIZABLE FRACTIONAL STABLE MOTION 361
instead. By Theorems 11.3.2 and 11.4.1 below, the process {Y(l,b;t), —oo <
t < oo} is well defined and has the following integral representation:
-oo
/■oo
/CO rU u ~U+IUX t ,
e«* f e—Zl du] (x~H+l-l/a + bxZH+i-l/a)M(dx)
-oo '-J— oo *•*• ^
r fi«x_iT_( -«+l-l/a + te-^+'->/«)M(^).
./-oo 1 + IX
-co
Its bivariate characteristic function
Eexp{i(zlY(hb;tl)+z2Y(l,b;t2))}
at ti = 0, *2 = t > 0, zi = 1, Z2 = —i equals exp{-co5(£>;£)}, where
B(6;t) = ||y(l,fc;0)+*y(l,6;t)||S
/•OO
= / {|l+ieixt|Q + 6Q|l+te-ixt|a}x-Q"+a-1(l+a:2)"a/2rfa;
Jo
/■oo
= 2Q/2 / {(l-sina;i)Q/2 + 6a(l+sina;i)Q/2}a;-QH+Q-1(l+a;2)-a/2da;.
Jo
The relation (7.8.5) implies
(1 +ba)-[B(b;t) = (1 +6'a)-1B(6';i)
or
b'a - ba f n _ 0
for all £ > 0, where
/»CO
/({) = / {(1 +Sina;i)a/2 - (1 - smxt)a/2}x-aH+a-\\ + x2)-a/2dx.
Jo
(7.8.7)
Since the following lemma shows that this is possible only if b — b', the proof is
complete. I
Lemma 7.8.3 Let 0 < a < 2, 0 < H < 1 and let f(t) be defined as in (7.8.7).
Then /(f) ^ Ofor some f > 0.
Proof: Suppose t ± 0 and consider firstly the case 1 < aH < 2. Then /(£)
equals
0rir/2t roo
' +/ ){(l+smxt)a'2-{l-smxt)a/2}x-aH+a-l(l+x2)-a/2dx
0 J-K/lt'

362
SELF-SIMILAR PROCESSES
7.8
r*/2t r / 2 \ <*/2 / 1 \ «/21
> I { (l + ±xt) - (l - ^xt) }x-aH+a~i(1 + x2)-Q/2dx
/•OO
-2/ x-^+^-Hl+x2)-^2^
J-jr/lt
since sinx > £x forO < x < 7r/2 and since (1 + sintx)"/2 - (1 - sinix)"/2 >
0 - 2"/2 > -2. Now, for tx > 0, (1 + f x*)Q/2 - (1 - \xt)a'2 has a Taylor
series expansion with only positive terms and hence it is strictly greater than ^atx.
Using also the inequality -(1 + x2)~al2 > —xa for x > 1, we obtain, for small
enough t,
*y ft/It fOO
f{t) > -at / x~aH+a{\ + x2)~a'2dx - 2 / x-aH-ldx.
* JO Jn/2t
Note that the integrand in the first integral is integrable as x —> oo if aH > 1
because it behaves like x~aH for large x. If aH = 1, the integral behaves like
In 3j for small enough t. The second integral, on the other hand, is proportional
to taH. Letting c,-(-, ■), j > 1, denote positive constants depending only on the
parameters in the parentheses, we have for sufficiently small t > 0,
fit) > c, (a, H)t - c2(a, H)taH > 0 if 1< aH < 2,
and
/(*) > c3(a)f ln(l/i) - cAt > 0 ifaif=l.
This proves the lemma in the case 1 < aH < 2.
Suppose now 0 < aH < 1. By changing variables, we obtain
fit) = tttH / {(1 + sinx)a/2 - (1 - sinx)a/2}x-QH+a-1it2 +x2)-a/2dx
Jo
/•oo
= taH I {(1 + sinx)a/2 - (1 - sinx)a/2}x-aH-]dx
Jo
r°° r / f2\ —ct/2 n
+ taH {(l+sinx)Q/2-(l-sinx)a/2}[(l + ^J -l\X-aH-ldx
(7.8.8)
:=taHic5ia,H)+git)).
It is easy to verify that the first integral in (7.8.8) converges. Indeed, its integrand
is integrable asi-»0 because it is asymptotically proportional to xx~aH~x and
aH < 1; it is also integrable as x —> oo because it is bounded by Ax~aH~x and
aH > 0. The integral is therefore equal to a constant, which we denote Cs(a, H).
We now want to show that
c5ia,H)>0.

7.9
SUBORDINATED PROCESSES
363
Write J0°° = 53fclo /2Jfc * anc* change variables to obtain
c3(a,H)= / {(l+sinu)a/2-(l-sinu)Q/2}V(u + 27rfc)-Q'ff-1du =
/•7T 0°
/{(l+sinu)Q/2-(l-sinu)a/2}y;((u+27rA:)-QH-1-(u+27rfc+7r)-QH-1)du.
J° fc=o
Since a;~ai?_1 is strictly decreasing on (0, oo), we conclude that c$(a, H) > 0.
We now turn to the second integral in (7.8.8) which we denoted g(t). Choosing
e > 0 and t > 0 sufficiently small, we see that \g(t)\ is bounded above by
/•OO
/ 1(1 + sina;)Q/2 - (1 - sinz)a/2| |1 - (1 + t2/x2)~a/2\x-aH-ldx
Jo
re /»oo
< 2 / x~aHdx + 2 (t2/x2)x-aH-
raH~xdx
/o
1-aH aH + 2
<Ci{a,H).
Using (7.8.8), we conclude that f(t) > 0 for sufficiently small t > 0. 1
7.9 Subordinated processes
A subordinated process X(t) is an a-stable process with representation
X{t)= I Y(t,x)M(dx), (7.9.1)
where Y = {Y(t, x), t G T, x £ £2} is a stochastic process defined on (Q, .T7, P)
and M is an a-stable random measure with control measure P.
For example, suppose that M is SaS, 0 < a < 2, and Y is Gaussian. Then X
is the SaS sub-Gaussian process introduced in Section 3.7. Suppose, now, that Y
is Sa'S with 0 < a < a' < 2. Then X is the SaS sub-stable process introduced
in Section 3.8.
The measure M in (7.9.1) does not have to be symmetric nor the process Y
be Gaussian or stable. In order to ensure that the process X is well defined, it is
only necessary to suppose E\Y(t)\a < oo for all t € T.
For example, if M is symmetric, then
d . d
£{expi(^^X(ij))} = exp{- / l^SjYitj^TPidx)},

364 SELF-SIMILAR PROCESSES 7.9
that is,
a a
EJexpi^^A-te))} =exp{-iiH j>nti)r}, (7-9.2)
where £p is the expectation with respect to the probability measure P.
If M is not symmetric, then one has also to take into account the skewness
intensity /?(•). In this context, /?(•) is a random variable because it is a measurable
function on (Q,!F,P). For convenience, suppose a ^ 1. Then the finite-
dimensional characteristic function of X(t) is
. d d
exp{-/ l£W*;,*)r(l-^(z)sign (5>;n*;.*)tan^))P(da:)}.
If /3 and V, which are both defined on {Q.,T, P), are independent, then the
finite-dimensional characteristic function of X(t) simplifies to
exp^EplY.OjY^T-iiEp^Ep^OjYitj)) * tan™},
j=i j=i
so assuming /3 and Y independent is equivalent to assuming that /3 is constant.
For convenience, we will suppose in the sequel that M is symmetric, so that
the finite-dimensional characteristic function of X(t) is given by (7.9.2). Observe
that the finite-dimensional distributions of X depend on those of Y only through
the a-moments of linear combinations of Y(t)s. Thus, if two different processes
[Yi(t), t 6 T} and {Y2(t), t G T}, the first defined on (Qi^Pi), the second
on (Cl2,3:2,P2), satisfy
d a d a
^,|X>*i(*j)| = EpJfcOjYiiWl
i=\ i=i
for all d > 1, 6\,.--,0d real, t\,..., td G T, then the processes
{*,(*) = J Yl(t,x)MPl(dx), t e T}
and
{X2(t) = J Y2(t,x)Mp2(dx), t € T}
have identical finite-dimensional distributions, where Mp, and Mp2 are SaS
random measures with control measures Pi and P2, respectively.
We now examine what happens if Y is an H-sssi process.

7.9
SUBORDINATED PROCESSES
365
Theorem 7.9.1 If{Y(t), —oo < t < oo} is H-sssi with finite a-moments defined
on (Q,, T, P) and if M is a SaS random measure with control measure P, then
X(t)= [ Y{t,x)M(dx)
Jn
is SaS and also H-sssi.
Proof: To verify self-similarity, let a > 0, and observe that
d d
£exp{i]T0jA'(at,-)} = exp{-£| ^(^(at,)!"}
d d
= exv{-aaH E\Y,9jY(tj)\a] = EexpU^Oj^X^Y
Verifying stationarity of the increments is also straightforward. I
Observe that if a = 2, then whatever the choice of the if-sssi process Y,
the resulting subordinated process X is always fractional Brownian motion (see
Definition 7.2.2). We now present several examples with 0 < a < 2 and M SaS.
Example 7.9.2 Suppose Y is fractional Brownian motion (a' = 2). The
corresponding subordinated process X is called sub-FBM. Its parameters are
0 6(0,2), He (0,1).
Because the process X is sub-Gaussian, it is different from the real harmonizable
fractional stable motion (Theorem 6.6.5) and from the linear fractional stable
motion (Theorem 4.7.5).
Example 7.9.3 Suppose Y is the linear fractional stable motion (LFSM) with
q' < 2 and take a < a'. The corresponding process X is sub-stable with
parameters
a €(0,2), a'e(a,2), IT e (0,1), a,6eR.
Example 7.9.4 Suppose Y is the log-fractional stable motion (logFSM) with
1 < a' < 2 (and H = I/a'). The corresponding process X is sub-stable with
parameters
a6(0,2), a'e(max(l,a),2), H=l/a'.
Example 7.9.5 Suppose Y is the real harmonizable fractional stable motion
(RHFSM) with a' < 2 and take a < a'. The corresponding process X is
sub-stable with parameters
ae(0,2), a'e(a,2), H € (0,1).

366
SELF-SIMILAR PROCESSES
7.10
Example 7.9.6 Suppose that Y(t) is a Hermite process (see Taqqu (1979), where
the process is denoted Zm(t)). Formally,
{y(t) = r dBfo) [ ' dB(&) • • ■ f m_' dB(u)
^ J—CO J — OO J —CO
rt m
x / TT(s-&)Hfl-3/2i(& <*)<*«, *^°}.
where B is a Gaussian random measure with Lebesgue control measure, m is
a positive integer and 1 - l/(2m) < H0 < 1. The process {Y(t), £ > 0}
is iJ-sssi with if = m(Ho — 1) + 1. It has moments of all order. When
m = 1, y(t) becomes fractional Brownian motion with 1/2 < H = Hq < 1.
When to = 2,3,..., F(t) is non-Gaussian.
The parameters of the corresponding process X are
a €(0,2), m= 1,2,3,..., H £ (0,1).
See Section 12.4 for more examples.
7.10 Fractional stable noises
If {X(t), t € R} is a process with stationary increments, then its increments
Yj = X(j + 1) - X(j), j = ..., —1,0,1,..., form a stationary sequence.
The sequence {Yj} is called a noise. "Fractional noises" are obtained by taking
the increments of H-sssi processes. For example, the fractional Gaussian noise
denned in Section 7.2.3 is the increment of the iJ-sssi Gaussian process fractional
Brownian motion. But although there is only one if-sssi Gaussian process, there
are many different //-sssi a-stable processes with a given 0 < a < 2, and each
of these processes gives rise to a different fractional a-stable noise.
Consider, for example, the linear fractional stable motion LQ,H(a, b; t)
introduced in Section 7.4. The linear fractional stable noise is the stationary sequence
Yj - La<H{o;b;j + \) - LQtH(a,b;j)
= rfaHj + l-xfi-t-ti-x)?--]
J —OO ^
+ b[(j + 1 - x)"~' - (j - x)"~-])M(dx), (7.10.1)
where M is an a-stable random measure with Lebesgue control measure.
The integrand in (7.10.1) is similar to that in (7.2.18) with H — \/a replacing
H — 1/2. By analogy, we will say that the YjS display long-range dependence

7.10
FRACTIONAL STABLE NOISES
367
when H > 1/q and negative dependence when H < 1/a. Because H is still
constrained to lie in the interval (0,1), long-range dependence is only possible
when a > 1. This fact can be understood heuristically as follows. When a < 1,
M generates many large values. If, in addition, H > 1/q, then their effect is long
lasting because the integrand in (7.10.1) changes very slowly with x. As a result,
the integral diverges.
We now list, for reference purposes, some other important fractional stable
noises. The log-fractional stable noise is
Yj = AQ,i/aO'+ 1) -AQiI/a(j)
/oo
(ln\j+l-x\-\n\j-x\)M(dx),
-oo
where M is a-stable with Lebesgue control measure. It is defined for 1 < a < 2.
The real harmonizable fractional stable noise is
Yj = HaM(j + \)-HaiHU)
ei*3*--Zl\x\-H+\-l/c.M(dx)t
IX
where 0 < a < 2, 0 < H < 1 and where M is a complex isotropic SaS random
measure with Lebesgue control measure.
Finally, the sub-Gaussian fractional stable noise is
Yj = Al'2(BH(j + 1) - BhU)) = Al/2Yji2,
where Bh is fractional Brownian motion, Yjj2 fractional Gaussian noise and A is
an independent ^-stable random variable defined in (2.5.1).
Additional fractional stable noises can be obtained by taking increments of
other if-sssi subordinated process; for example, those defined in see Section 7.9.
The asymptotic dependence structure of the fractional Gaussian noise {Y(j)}
was investigated in Proposition 7.2.10 where we noted that the autocovariance
function EY(j)Y(0) is asymptotically proportional to j2H~2 as j —> oo. Here, we
shall use the codifference r(j) to analyze the asymptotic dependence of fractional
stable noises.
We consider only the linear fractional stable noise and the log-fractional stable
noise because the codifference does not tend to zero for either the real
harmonizable fractional stable noise or for the sub-Gaussian fractional stable noise (see
Propositions 6.7.1 and 4.7.4). The first two types of noises are stationary moving
averages and, therefore, their codifference t(J) tends to 0 as the lag j —► oo
(Theorem 4.7.3).
- */_

368
SELF-SIMILAR PROCESSES
7.10
The following theorems specify the asymptotic behavior of I(Q\, 62;j) defined
in Relation (4.7.5) for any 6ud2 € K and, hence, of r(j) = -1(1, -l;j). The
proofs, which are very technical, are omitted. They can be found in Astrauskas,
Levy and Taqqu (1991).
Theorem 7.10.1 Consider the linear fractional stable noise defined in Relation
(7.10.1) and suppose that it is SaS. If either
(i) 0<a< 1,0<H < 1
or
(ii) 1< o < 2, 1 - ;^L_ <H<l,H^l/a
then,9 as j —► oo,
i(eue2-j) ~ B(ei,e2)jaH-a,
t(j) ~ -B(\,-l)jaH-
then, as t —» oo
(Hi) Ka<2,0<H<l- , ' ...
1 ' ' a(a —1)'
/(0,,02;j) ~ F(dl,92)jH-^-\
r(j) F(l,-l)jH~^-K
The constant B(0\, 02) equals
H-1-
a
a\a f {^(l-a^-i-'+^-x)"-^
J — oo *■
H~i-l\a}dx
-OO
xH-i-lia
|0,(1 -x)W-o-'r-|^(_X)
»1
+y {p,(i-x)H-
°~' + 602a;H"
H-i-Ha
|a0i(l -x)
tf-i-lia
+ |6|Q / {\02(l+x)H-±-l + 01xIi-±-i\
- |02(1 +x)if-i-1|Q - Ifl.a^-i-'rjdx
9If a = 1, the asymptotic behavior of / is valid under the following additional conditions: either
sign ab = 1 or sign B\ 02 — — 1 ■ If & = 1 and both sign ab ^ 1 and sign 8\ 82 ^ — 1, then B = 0.
In that case one has to consider the next term in the asymptotic expansion and one finds that
i(e,,er,j)~C(elto2)jH-2
as j —► 00, where
C(0,, 02) = -2(1 - tf)(|ofl, I + |W2|).
(Recall: sign u ^ 1 means u < 0.)

7.10
FRACTIONAL STABLE NOISES
369
The constant F(Q\, B2) equals
(Ha - 1)
/ „ 1 „ 1 \<a-l>
(a02[(l - x)H-° - (-x)H~°]J dx
+ I" (d2[a(l - x)H~± -bxH-±]ya~X>dx
+ [°° (b62[(x - l)H~i - xH~±}ya~l>dx\
+ bejf (w,[(i - x)H-± - (-x)H~^ya~l>dx
+ f (eMl-xf-i -axH~±]y
/<x>
(a0,[(x - 1)H-^ - xH~i})
<Q-1>
<£»-!>
PROOF: See Astrauskas, Levy and Taqqu (1991), Theorems 2.1,2.2 and 2.3.
Theorem 7.10.2 For the log-fractional noise, as j —► 00,
WM) ~ G(9i,02)jl-a,
r(j) ~ -0(1,-1)^-°,
where
G(eue2) = / {
7— 00 ^
'1
+
/
Jo
°1
1 + X X
■CO
1 +X
02 <n_
\ + x x
1 +x
dx
dx
PROOF: See Astrauskas, Levy and Taqqu (1991), Theorem 2.4. I
The codifference r equals the covariance when a = 2 (Property 2.10.3).
Thus, for fractional Gaussian noise, t(j') « j2H~2 as j —> 00, where « denotes
asymptotic proportionality (compare with Theorem 7.2.11). Theorem 7.10.1 states
that in most cases r(j) « jaH~a when 0 < a < 2, with the following exception:
t(j) « j"_(1/of)_1, when a > 1 and 0 < H < H~, where
H = H(a) = 1
a(a — 1)'

370 SELF-SIMILAR PROCESSES 7.11
This phase transition 10 at H = H is peculiar. Since H(a) < 1/a, it occurs only
when there is negative dependence.
Note that H > 0 if and only if a > a0 where
00 = 1^ = 1.618,..
equals the golden ratio." Thus, r(j) « jH~(l/a)-1 takes place only when
1 < aQ < a < 2 and 0 < H < ~H{a),
that is, for large a and small H. In this region,
aH -a < H 1<0.
a
Finally, observe that the two exponents aH - a and H - (1 /a) - 1 are equal on
the boundary H = H(a).
Remarks
1. We have assumed that the noises are SaS. If they are skewed a-stable, the
asymptotic behavior of / and r is the same but the constants B;F and G are
more complicated. They are given in Astrauskas, Levy and Taqqu (1991).
2. If X is an iJ-sssi process, one can also define Y(t) = X(t + 1) — X{t) for
all t e M and not just for t = ...,— 1,0,1,..., as we have done in this
section. Theorems 7.10.1 and 7.10.2 continue to hold with j replaced by
tei.
3. The parameters 6\ and Qi affect only the multiplicative constants in the limit
and not the exponents.
7.11 Simulation of fractional noises and motions
To simulate fractional Brownian or stable noise (or motion) one approximates
the integral representation. This creates two types of errors, a "low frequency"
one due to the truncation of the limits of integration and a "high frequency" error
caused by replacing the integral by a sum. Consider then the fractional Gaussian
,0It does not occur, for example, when the process is the projection of a linear fractional random
field of dimension greater than one (Kokoszka & Taqqu 1993a).
1' Private communication of Jonathan Taqqu. The golden ratio a/b satisfies the relation a/b =
(a + b)/a.

7.11 SIMULATION OF FRACTIONAL NOISES AND MOTIONS 371
noise as defined in (7.2.18) and linear fractional stable noise with a = 1, 6 = 0,
given in (7.10.1). We simulate them as follows:
Mm
Yj = Z2 Kd(~)ei-(u/rn), 3 = 1> • • • ,T
(7.11.1)
u=\
where the es are i.i.d. a-stable random variables, 0 < a < 2. M and m are
integers. M defines the memory (cutoff in the limits of integration), 1/m is the
mesh (discretization of the integral) and T < M is the length of the sequence.
The kernel
( xd -(x- l)d, x> 1,
Kd(x) = I
[ xd, 0 < x < 1
with d = H - ~ is plotted in Figure 7.11 for d = ±0.4.
d = 0.4
Figure 7.11: The kernel Kd{x)
This procedure requires generating m(T + M — 1) i.i.d. e^s. We avoided infinite
values for d < 0 by starting the summation in (7.11.1) at 1/m and taking Kd(l)
equal to 1. It is best to choose a large value of m when d is negative. It is also
convenient to renormalize the sequence {Yj, j — 1,..., T} by subtracting the
sample mean and dividing by the sample standard deviation. To approximate
fractional Brownian or stable motion, one can use T~H J]j=i ^j> V^ < * < 1>
where [ ] denotes the integer part. For display purposes, however, it is often
sufficient to plot 5Dj-=i Y:> l = *> • • • >T with m adequate aspect ratio. The
following figures illustrate this procedure in the case a = 2 and a = 1.7. The
random variables £js are symmetric. We chose T = M = 1000 and m = 30
and focused on d = 0.4, 0, -0.4. Since H = d+l/a, this choice of d means
H = 0.9, 0.5, 0.1 in the Gaussian case and H = 0.99, 1/1.7 = 0.59, 0.19

FRACTIONAL GAUSSIAN NOISE
d=-0.4, H=0.1
400 600
d= 0.4, H=0.9
en
r
r
>
•a
50
O
o

7.11
SIMULATION OF FRACTIONAL NOISES AND MOTIONS
373
2
O
1-
o
o
tr
CQ
O
i-
o
<
GC

LINEAR FRACTIONAL STABLE NOISE, ALPHA=1.7
d= -0.4, H=0.19 d= -0.4, H=0.19, large scale
200 400 600 800
i.i.d. S1.7S, H=1/alpha=.59
0 200 400 600 800 1000
i.i.d. S1.7S, H=1/alpha=.59, large scale
^4^t^
200 400 600 800 1000
d=0.4, H=0.99
200 400 600 800 1000
d=0.4, H=0.99, large scale
^4¥^f^^
C/5
r
-n
CO
r
>
TO
TJ
JO
o
o
tn
in
200 400 600 800 1000
200 400 600 800 1000

LINEAR FRACTIONAL STABLE MOTION,
d=-0.4, H=0.19 i.i.d S1.7S, H=1 /alpha=0.59
ALPHA=1.7
d=0.4, H=0.99
&' A 1
*• \ i *? i-
Is * *
7 *
>■
i
i
•
*
•• »
it i ;
ft "'• *
tt7
5
.»'
?,
t
o
z
o
>
o
d
o
z
>
r
z
o
C3
m
on
>
Z
o
S
o
H
O
z
0 200 400 600 800 1000 0 200 400 600 800 1000
200 400 600 800 1000

376
SELF-SIMILAR PROCESSES
7.12
in the 51.75 case. The case d = 0 is included for reference and is obtained by
simply setting Yj = e,. The corresponding cumulative sum simulates Brownian
motion (a — 2) and stable LeVy motion (a = 1.7). Since the sample paths of
the stable LeVy motion are discontinuous, we have not connected the points in the
corresponding figure.
The smaller the value of d, the higher the fluctuations (high frequencies) of the
corresponding noises. Because of the scale, the high frequencies are not visible
in the figures displaying fractional Brownian or stable motion at d = 0.4. Long-
range dependence affects the low frequencies. Their presence is quite noticeable
in the figures displaying the noise when d — 0.4.
7.12 ARM A sequences with stable innovations
Autoregressive-moving average (ARMA) processes are often used for modeling
empirical time series. Let p and q be non-negative integers. The sequence
{Xn, n = ..., —1,0,1,...} is called ARMA (p, q) if it satisfies the equations
X„ - <t>\Xn-\ (j>pXn^p = en + 6\en-i -) 1- 0qen-q. (7.12.1)
The innovations en are i.i.d. random variables. In the classical time series
literature, the ens are either Gaussian or non-Gaussian with finite variance and therefore
the probability that they take large values is very small.
Here, we shall suppose that the ens are i.i.d. a-stable with 0 < a < 2; in
other words, en ~ Sa(a,f3,fi) if 0 < a < 2 and N(fi,2a2) if a = 2. The
finite-dimensional distributions of the Xns depend on the coefficients 6\,..., 6P
and <f>},..., <pq.
Consider the system (7.12.1) with real coefficients 4>o = 15 4>i > • • • > <t>pi &o —
1, B\,...,0q and define the polynomials
<D(z) = \-faz 4>Pzp,
e(z) = i + d]Z + ■■■ + eqz<>,
where z is a complex variable. One can write (7.12.1) symbolically as
<D(£)X„ = e(J3)£„, n = .... -1,0,1,..., (7.12.2)
where B is the backward operator, formally, BiX^) = Xn-\, B2(Xn) =
Xn-2, • • ■ • As in the Gaussian case, one solves (7.12.T)*by showing that Xn =
(&(B)~1Q(B)en is well defined. It is natural to suppose
Condition 7.12.1 The polynomials Q(z) and <S>(z) do not have common roots.

7.12
ARMA SEQUENCES WITH STABLE INNOVATIONS
377
The following theorem shows that, as in the Gaussian case, a non-anticipating
solution exists if and only if(^(z) has no roots in the closed unit disk {z : \z\ < 1}.
Theorem 7.12.2 The system (7.12.1) has a unique solution of the form
oo
Xn = ^Cj£n_j, neZ, a.s. (7.12.3)
j=o
with real CjS satisfying \cj\ < Q~j eventually,12 Q > 1, if and only (/O(z) has
no roots in the closed unit disk {z: \z\ < 1}. The sequence {Xn, n € Z} is then
stationary and a-stable.
The CjS are the coefficients in the series expansion o/0(z)/O(z), \z\ < 1.
PROOF: Suppose O(z) has no roots in {z: \z\ < 1}. The function
0(z)
C(z)
<b(z)
is, therefore, analytic in the disk {z: \z\ < R}, where R > 1 is the radius of
convergence of the series C(z) — Y^T=qcjz^ • Since \/R = limsup,^^ lc-,11^,
for any 1 < Q < R, \cj\ < Q~j eventually. Using the relation Q>(z)C(z) =
8(z), which holds for \z\ < 1, and the fact that the series Y^jLa C5Z^ converges
absolutely for \z\ < 1, we obtain the following system of equations:
CO
C\
02
= I,
- 0iCo
-<$>\C\
= 01,
- <^2C0 =
(7.12.4)
Cq - 0lCq_i - <h.Cq-l ~ ■■■ ~ 4>qCQ = 0q,
Cs - <t>\Cs-\ - 02CS_2 - ... - <S>sCq - 0, S > q,
with the understanding that <fo = 0 if i > p. It follows from (7.12.4) that the CjS
are real and since \cj\ < Q~j eventually, the series (7.12.3) is well defined and,
in fact, converges absolutely a.s. (Exercise 1.26).
To see that the process (7.12.3) with the CjS uniquely defined by (7.12.4)
satisfies (7.12.1), just use Relation (7.12.4) and the fact that the series (7.12.3)
converges absolutely a.s.. Rearranging the terms in
oo oc oo
j=0 j=0 3=0
yields (7.12.1).
I2"aj < bj eventually" means that there is a j0 such that a.j < bj for all j > jo-

378
SELF-SIMILAR PROCESSES
7.12
To prove the converse suppose that the system of equations (7.12.1) has a
solution of the form (7.12.3) with the c,s satisfying \cj\ < Q~i eventually for
some Q > 1. We want to show that O(z) ^ 0 for \z\ < 1.
Consider the series C(z) = Yl7LocizJ which, under our assumptions,
converges absolutely and uniformly in the closed unit disk {z: \z\ < 1}. Setting
oo
e(z):=®(z)C(z)=:Y/6jz\ \z\<\, (7.12.5)
we obtain
#o = Co,
(7.12.6)
®s■— cs - 4>\cs-i 4>scq , s > 1.
Since, for any n, the series X!jloci£n-j converges absolutely a.s., (7.12.1) and
(7.12.6) imply
q oo
2j#,en_j = Xn — 4>\Xn-\ (f>pXn_p = 2J#.,en_j a.s., (7.12.7)
j=0 j=0
which, in turn, yields 6j — 63; for j = 0,1,..., q and 6j = 0 for j > q. Thus,
e(z) = Q(z) and, by (7.12.5),
•W-^. MSI-
As C(z) is bounded on {z: \z\ < 1}, 3>(z) = 0 implies Q(z) = 0. But O(z)
and Q(z) do not have common roots, so <t(z) ^ 0, for all \z\ < 1, proving the
converse.
The solution (7.12.3) is a-stable because it is a linear combination of a-stable
random variables. It is clearly stationary. I
The condition that <l>(z) has no roots in the closed unit disk {z: \z\ < 1} is
a natural one for it ensures that the system of equations (7.12.1) has the
stationary non-anticipating solution (7.12.3). We shall suppose from now on that this
condition holds.
The CjS are obtained by identifying the coefficients of C(z) = Y?jLoc3z*
with those in the power series expansion of Q(z)/Q>(z). Since this is the same
procedure as in the Gaussian case, the explicit form of the CjS for specific ARM A
(p, q) models can be readily found in the time series literature.
Example 7.12.3 Consider the autoregressive process {Xn} of order 2 defined by
Xn - <j>\Xn-\ - faX-i = €n.

7.12
ARMA SEQUENCES WITH STABLE INNOVATIONS
379
If 4>(z) = 1 — 4>\z — 4>2Z2 has two different roots z\ and z2, satisfying \zi\ >
1, i = 1,2, then
®(z) = -±-(z-z1)(z-z2)
Z\Z2
and
-i „-i
1 Z\Z2 I \ \ \ Z\Z2 ( Z, Z.
/_1 1 \ _ zxz2 / z, z2 \
0(z) z2- z\\z\- z z-i — z) z2- z\\\ -(z/zi) \-{z/z2))'
The coefficients Cj in the series expansion of l/®(z) are, therefore,
" = ^(^'-^-').^o.
If <&(z) has complex conjugate roots pe±tM, \i ^ kir, then it is not difficult to see
that
sin/i(j + l) .
Cj = : p J.
sin/i
The ARMA (p, q) time series is invertible if there exists a sequence of
constants {cj} such that ^Zjlo l^'l < °° and Z)^o?j'-^n-i = €™> n 6 ^' wnere
convergence holds in probability. Invertibility is particularly useful for prediction
because it allows Xn to be expressed in terms of the previous observations Xj, j <
n. The following theorem provides a condition for invertibility.
Theorem 7.12.4 Suppose that Q{z) has no roots in the closed unit disk {z: \z\ <
1}. Then ARMA (p,q) is invertible, i.e.,
OO
^2cjXn-j = en, n e Z, a.s..
j=o
77ze Cjj are f/ie coefficients in the series expansion ofQ~x {z)Q(z), \z\ < 1.
PROOF: Because 6~'(z)0(z) is analytic in the disk {z: \z\ < R}, R > 1, we
have Y^'jLo |cj | < oo. If a > 1, we have E\Xn \ < oo and the result follows from
Exercise 7.17. In the general case 0 < a < 2, it follows from Exercise 7.18. I
Since the coefficients c,- of the moving average (7.12.3) satisfy \cj\ < Q_J
eventually with Q > 1, they lie within two exponentially decreasing functions.
This behavior is quite different from that of either fractional stable noise or of the
fractional ARIMA considered in the next section. In these cases, Cj decreases like
a power function.
As for the codifference t(ti) = Txn,x<,> we have:

380 SELF-SIMILAR PROCESSES 7.13
Theorem 7.12.5 Suppose that <t>(z) has no roots in the closed unit disk {z: \z\ <
1} and let Q > 1 be as in Theorem 7.122. Then there are constants K\ and Ki
depending ona,Q and the CjS such that
limsup,,^Q"|r(n)| < Kx forl<a<2,
limsupn_00(3a"|r(n)| < K2 for0<a<l.
PROOF: See Kokoszka and Taqqu (1993d). I
If a > 1, a similar result holds for the covariation, namely
limsupQn|[A;,,*0]Q|<if3,
n—+oo
because
oo
i=0
oo oo
< const. YJQnQ~U+n)Q~{a~X)J = const. ^Q-QJ' < oo.
7.13 Fractional ARIMA with stable innovations
We now turn to the fractional ARIMA time series, also called FARIMA or
fractional ARMA. "ARIMA" stands for "autoregressive integrated moving average."
Let A be the difference operator, defined by AXn = Xn — Xn_i = (I — B)Xn
and let Ad, for d = 1,2,..., be A iterated d times. The fractional ARIMA model
is based on the classical ARIMA (p, d, q) model
®(B)AdXn - Q(B)en,
but the parameter d, instead of being a non-negative integer, is allowed to take
fractional values, either positive or negative. To simplify the discussion, set
p = q = 0. If d is a non-negative integer, then AdXn = en describes a model
where Xn, differenced d times, yields a sequence of i.i.d. random variables (the
random walk, for example, is ARIMA (0,1,0)).
Now define AdXn = en for d fractional as Xn = A~den and interpret
A~d = (I - B)~d by using the formal power series expansion (1 — z)~d =
££Lo bj\-d)zi as follows:
OO
j=0

7.13
FRACTIONAL ARIMA WITH STABLE INNOVATIONS
381
where B^ denotes the backward operator B iterated j times. The coefficients
bj(—d) in the expansion are bo(—d) = 1 and
6.M) = ^___ = ______ J = lj2i...) (7.13.1)
where T denotes the gamma function: r(x + 1) = xT(x) and T(j + 1) = j\ for
j integer.
The formal definition of fractional ARIMA (0, d, 0) is, then, as follows: it is
the moving average
oo
Xn = J2bj(-d)£"-J' « = •••.-!.0,1,-.., (7.13.2)
j=0
where the bjS are given by (7.13.1).
Note that for d = 0,-1, —2,..., the operator &~d is merely the difference
operator A iterated \d\ times, and in this case,
bj(-d) = 0 for j > d if d = 0,-1,-2,....
For example, Xn — en when d = 0, Xn = en — en_i when d = — 1 and
-Xn = £n - 2e„_i + «„_2 when d = —1.
Applying Stirling's formula to (7.13.1), we obtain
ft^-d)-^"-1 asj-»oo if d± 0,-1,-2,.... (7.13.3)
Now suppose en ~ 5Q(cr,/3,/i). The following theorem specifies the
conditions on a, d and /i under which fractional ARIMA (0, d, 0) is defined.
Theorem 7.13.1 Let en ~ Sa (a, f3,y) be i.i.d. and suppose
-oo<d<\--. (7.13.4)
a
Condition (7.13.4) is necessary for the series (7.13.2) to converge. When it holds,
the series (7.13.2) converges in the following sense:
(1) 0 < a < 1: absolutely a.s.;
(2) 1 < a < 2: absolutely a.s. ifd<0, and a.s. ifd > 0 and p,-0.
Proof: Observe firstly that TT=j0Ud~l)a < °°. Jo > L if md only if d <
1 - i. Relation (7.13.3) and Exercise 1.26 imply that the series (7.13.2) converges
absolutely a.s. in the cases indicated. When 1 < a < 2 and d > 0, one does not

382
SELF-SIMILAR PROCESSES
7.13
0.5 -
(2,0.5)
Figure 7.12: The shaded region indicates the
allowable values of (d, a) for fractional ARIMA.
have J2JLq \bj(-d)\ < oo. However, £°10 \bj(-d)\a < oo and ^ = 0 imply
that the series (7.13.2) converges a.s. I
Figure 7.12 illustrates the region of the (d, a) plane where Condition (7.13.4)
is satisfied.
Both fractional stable noise and fractional ARIMA (0, d, 0) are moving
averages with coefficients that decrease like a power function. It is tempting to
compare the asymptotic rate of decay of the kernels. For fractional stable noise
(j + 1 - x)*-'/Q - (j - x)H-"» « (j - a;)"-(>/«)-i as j - x -> oo and
for fractional ARIMA (0,d,0), bj(-d)
following relation between H and d:
..d-i
as j —► oo. This suggests the
d=H
a
(7.13.5)
As for fractional stable noise we will say that d > 0 corresponds to long-range
dependence and d < 0 corresponds to negative dependence. Since d < 1 — 1/a,
long-range dependence can occur only if a > 1.
Recall that fractional stable noise is defined only for 0 < H < 1, whereas
fractional ARIMA (0, d, 0) is defined for all d < 1 - 1/a. The condition H < 1
is equivalent to d < 1 — 1 /a. The lower bound H > 0, expressed in terms of

7.13
FRACTIONAL ARIMA WITH STABLE INNOVATIONS
383
d = H — 1/a, becomes d > —-. Thus 0 < H < 1 becomes
-!<d<i-I,
a a
an important range for fractional ARIMA.
Indeed, suppose d < 1 - 1/a. Since A~d — AmA~d , where m is a non-
negative integer and -1/a < d' < 1 - 1/a, the operator &~d with d < 1 - 1/a
can be interpreted as A-d followed by a number of full differences (Exercise
7.16).
We have been supposing until now p = q = 0. It is straightforward to define
a general fractional ARIMA (p, d, q) with d < 1 — i, by
*(B)Xn = 0(£OA-den, (7.13.6)
or, formally,
Xn=<S>(B)-lQ(B)A-den, (7.13.7)
under the assumptions given in Section 7.12, namely Condition 7.12.1 and that
0(2) has no roots in the closed unit disk {z: \z\ < 1}. Relations (7.13.6) and
(7.13.7) are to be interpreted as
cc
Xn = J2ujtn-j, (7.13.8)
where
j
Uj = (c*b{-d))j =YJdh-i{-d), (7.13.9)
i=0
and where the coefficients Ci are those in (7.12.3). The UjS are thus the coefficients
in the power series expansion of Q>~[ (z)Q(z)(l — z)~d, \z\ < 1.
The effect of the operator O-' (B)B(B) is to provide modeling flexibility. It
modifies the coefficients of the moving average without modifying their
asymptotic behavior.
Combining Theorem 7.12.2 with the proof of Theorem 7.13.1, we obtain:
Theorem 7.13.2,Suppose that the polynomials <b(z) and Q{z) satisfy Condition
7.12.1 and that <£>(z) has no roots in the closed unit disk {z: \z\ < 1}. Suppose
also that
-co < d < 1 ,
a
and en ~ Sa{o,(3,p) i.i.d. Then the series (7.13.8) defining fractional ARIMA
(p, d, q) converges in the following sense:
(1) 0 < a < 1: absolutely a.s.;
(2) 1 < a < 2: absolutely a.s. if d < 0, and a.s. ifd > 0 and p, — 0.
It is a solution of the system of equations (7.13.6).

384 SELF-SIMILAR PROCESSES 7.13
PROOF: Use Exercises 7.15 and 7.18. ■
We now address the question of invertibility of fractional ARIMA (p, d, q)
when a > 1 (see Figure 7.13).
0.5 -
-0.5 -
-l
d= 1 - \/a
IN.
<*=-(1-1/q)
(2,0.5)
(2, -0.5)
Figure 7.13: Fractional ARIMA is invertible if
(d, a) lies in the shaded region.
Theorem 7.13.3 Suppose that Q(z) has no roots in the closed unit disk {z: \z\ <
1} and let Uj be the coefficients in the series expansion of©~!(z)<$>(z)(l — z)d.
\d\<\ l
a
then
Y^UjXn-j = en, n € Z,
3=0
where the convergence is in IP, 0 < p < a. For 0 < d < 1 — I/a, the partial
sums J2T=o UjXn-j converges to tn absolutely a.s.
PROOF: Use Exercise 7.18. I
The following theorem gives the asymptotic behavior of the codifference
-r(n) = TXt+n,xt-
Theorem 7.13.4 Suppose Xn = Y?jLoujen-j is a fractional ARIMA (p,d,q)
process with SaS innovations en defined in (7.13.8). Suppose 0 < a < 2 and d
is not an integer.
(a) If either (i) a < 1 or (ii) a > 1 and (a - l)(d - 1) > -1, then
r(n)
lim . .
n-.oo na(d-I)+1
0(1)
r(d)*(0
l g(x)dx,
Jo
(7.13.10)

7.13
FRACTIONAL ARIMA WITH STABLE INNOVATIONS
385
where
g{x) = x^l)a + (1 + z)(d-1)Q ~{xd-x - (1 + x)d-l)a. (7.13.11)
(b)Ifa> land (a- \){d- 1) < -I, then
limiW=^egL J2 <„-!>. (71312)
n-^oo n^-' r(d)<E»(l) 4^ 3
(Ifd is an integer, then {Xn} is a finite moving average and, consequently, rn = 0
for large n.)
PROOF: See Kokoszka and Taqqu (1993c). I
Remarks. The condition (a- l)(d- 1) > — 1 can be rewritten as d > l-l/(a —
1). Figure 7.14 shows the regions of the (d, a) plane where the different rates are
in effect.13 The rates are the same as those for the linear fractional stable noise
(Theorem 7.10.1).14
Instead of the codifference one may consider the covariation [Xn,Xo]a of
Xn and Xo- This covariation, however, is defined only for q > 1. Since
Xn = Y^jLo uj£n-j, we have, by Proposition 3.5.2,
oo
[Xn,X0]a = YJui+nufa-x> ■ (7.13.13)
The next theorem shows that the asymptotic behavior of this covariation is similar
to that of the codifference.
Theorem 7.13.5 Suppose that {Xn} is a SaS fractional ARIMA(p, d, q) process
defined in (7.13.8), and that 1 < a < 2 and that d is not an integer,
(a) If (a- \)(d- 1) > -l.then
urn —,. ,. ,,
0(1)
r(d)*(i)
/ (y+i)d-ly{a-lKd-l)dy. (7.13.14)
Jo
(b) If {a- l)(d- 1) < -I, then
Xn,Xo]a _ 6(1) V„,«*-l> (1 1-3 lO
™ nV-D r(d)0(l)
13See the remark at the end of this section for the case a = 2.
14To compare the rates, assume that d > —1/a and set H = d — 1/a, d\ = 1 and 82 — —1
in Theorem 7.10.1. The assumption H ^ 1/a corresponds to d ^ 0. Cases (i) and (ii) of Theorem
7.10.1 correspond to case (a) of Theorem 7.13.4 and case (iii) corresponds to case (b).

386
SELF-SIMILAR PROCESSES
7.13
d= 1 - l/(a- 1)
Figure 7.14: Regions in the (d, a) plane where the various rates for
the codifference (and for the covariation, if a > 1) are in effect. (See
Theorems 7.13.4 and 7.13.5.) Regions A and B correspond,
respectively, to the rates nQ(d~l>+1 and nd~'.
PROOF: See Kokoszka and Taqqu (1993c). I
We conclude from these results that, for a > 1, the covariation [Xn,X0]Q and
codifference r(n) are asymptotically proportional. The constant of proportionality
depends only on a and d.
Remark. Theorems 7.13.4 and 7.13.5 and their proofs remain valid in the
Gaussian casea = 2. Notice that if a = 2andd < 0, Relations (7.13.12) and (7.13.15)
become, respectively,
r{n)
lim ,
n—>oo n
0
and
because for d < 0,
lim
\Xn, Xo]a
TV
d-\
j=0
0.

7.14
EXERCISES
387
This does not contradict the fact that for a = 2 the actual rate of decay is nld~x.
It is the special relation Y^oufa~l> = 0 vauc* f°r ot = 2, which accounts for
the discontinuity at a = 2 (and d < 0) in the rate of decay of the codifference
and the covariation of the fractional ARJMA (p, d, q) time series. Observe finally
that 2d — 1 = a(d — 1) + 1 when a = 2. Thus, in the Gaussian case a = 2,
the rate in part (a) of Theorem 7.13.4 is in effect not only for d > 0 but for all
-co < d < 1/2, as Figure 7.14 illustrates.
For more details on fractional ARIMA with stable innovations, see Kokoszka
andTaqqu (1993c).
7.14 Exercises
Exercise 7.1 Let X\ and X2 be two independent stochastic processes on R and
let
xloX2 = {xl(x2(t)),teR}
be the composition process. (Assume X\ measurable, i.e., the map (t,u>) i—>
X\ (t, w) is measurable, so that the composition process X\ o X2 is measurable.)
Show that if X\ and X2 are Hj-sssi, j = 1,2, then X\ o X2 is H\i?2-sssi.
Hint: Vervaat (1985).
Exercise 7.2 Prove that (7.2.7) represents standard fractional Brownian motion
up to a multiplicative constant. Determine that multiplicative constant.
Exercise 7.3 Prove that {/^(ln \t-x\- In \x\)M{dx), t E K}, is well defined
and that it represents standard Brownian motion up to a multiplicative constant.
Determine that multiplicative constant.
Exercise 7.4 Prove that for every 7 > 0 and complex z = x + iy, x > 0,
Jo zJ
where F is the Gamma function and where z~< denotes the branch obtained by
continuity for x > 0 starting from the positive real branch for z positive. Show
by continuity that this formula is still valid for x > 0, z ^ 0 if 0 < 7 < 1. Thus,
in particular
J/. 00
' eiuu'<-idu = ei^/2T{1).
0
Hint: See Choquet-Bruhat (1967).
Exercise 7.5 Make the passage from the representations (7.2.10) to (7.2.12)
rigorous.
Hint: See Section 6 of Taqqu (1979).

388 SELF-SIMILAR PROCESSES 7.14
Exercise 7.6 Show that the representation (7.2.12) defines an Jf-sssi process.
Exercise 7.7 Show that (7.2.17) represents standard fractional Brownian motion
up to a multiplicative constant. Determine that multiplicative constant.
Exercise 7.8 Let {Y,, j — ..., —1,0,1...} be fractional Gaussian noise. Use
its moving average representation (7.2.18) to compute its autocovariance r(j) =
EY0Yj.
Exercise 7.9 Prove Corollary 7.2.12.
Exercise 7.10 Let || • || be the Euclidean norm on Kn. For n > 2,0 < a < 2, 0 <
H < l.set
xn,a,H{t)= f (l|x-a||H-("/QM|x||"-(n/Q))M(dx), tern, (7.14.1)
JR"
where x = (xit... ,xn), l = (l,...,l)€Rn and M is a SaS random measure
on Rn, 0 < a < 2, with Lebesgue control measure. Show that the processes
Xn>a%H are well defined and iJ-sssi.
Exercise 7.11 Show that for any m, n > 2, m ^ n, 0 < a < 2, 0 < H < I,
the processes Xn,a,H and Xm,aH of Exercise 7.10 are different.
Hint: See Samorodnitsky and Taqqu (1990a), Theorem 3.1. In addition, make
the following corrections to page 312 of that paper: on line 7, replace n = 2 by
n = 2,3,4,5, and on line 9, replace n > 3 by n > 6.
Exercise 7.12 Show that the a-stable Levy motion in (7.5.1) and the log-fractional
a-stable motion defined in (7.6.1) are different from the process Xn<a,H defined in
Exercise 7.10 for any \<a<2, n>2, H= l/a. Observe that Xnta<1/a, n >
2, 1 < a < 2 provide new examples of 1/a-sssi processes.
Exercise 7.13 Prove that the real harmonizable fractional stable motion (7.7.1) is
i?-sssi.
Exercise 7.14 Let
/oo
eixtf(x)M{dx), <eR,
-oo
where M is a complex SaS, 0 < a < 2, random measure on R x 52 with circular
control measure mj where m is the Lebesgue measure on R and 7 is the uniform
probability measure on 52. Show that if one sets a = 2 in its joint characteristic
functions, the latter become that of
/oo
eixtf(x)M(dx), teR,
-OO

7.14
EXERCISES
389
where C is a constant and M is a complex Gaussian random measure with control
measure m defined as in Section 7.2.
Exercise 7.15 Prove that in Theorem 7.13.2, for d a non-negative integer,
j OO
lim j-^VCjbj-i^TWY,*.
=0 i=0
Hint: Divide the inner summation in two: 1 < i < j/2 and j/2 < i < j and
bound the terms accordingly.
Exercise 7.16 Let 0 < a < 2, d < 1 - 1/a, e,- i.i.d. SaS and x[d) =
X^Lo^'l-^)£t-ji * G Z, where 6j(—c() is defined in (7.13.1). Then for any
m = 0,l,2,...,
x(d-m) = (1 _ BjmX(d)) t e z_ (7.14.2)
//<«/; Let m — 1. Show lim^oo £>s(—d)et_j_s = 0 a.s. (Borel-Cantelli
lemma) and bj(-d) — bj-\(-d) = bj(-d — 1), j > 1. Then consider
(1 - B) lim^oo J2Sj=o bj{-d)et-j. (You could use Exercise 7.18 below.)
Exercise 7.17 Consider an arbitrary sequence of random variables {Xn}'%L_00
such that supni?|Xn| < oo. Show that if Yl'jL-oo \aj\ < °°> tnen *e series
Sji-oo ajXn-j converges absolutely a.s.
Hint: Proposition 3.1.1 of Brockwell and Davis (1991).
Exercise 7.18 Prove the following:
Suppose {en, n € Z} is a sequence of i.i.d. SaS random variables with
0 < a < 2. Let {cj, j — 0,1,2,...} be a sequence of real numbers satisfying
DjLo \ci\a < °° and let i^i' j = 0,1,2,...} be a sequence of real numbers
such that E°t0 W>j| < °o if a > 1, and E°l0 fe|a < oo if a < 1. Let
oo oo
Xn : = C{B)en = £Cie„_j and Yn : = ¥(B)en = J^^n-j,
and let A(B)en := Ejloaien-J- where ai = Ei=o^feci-*' i = 0,1,2,... .
Then
m oo
lim y^ipkXn-k = y^aj£n-j (7.14.3)
i—.oo *■—' z—J
m—>oo .
fc=0 J=0
and
!™, S cJy»-i= ^ aj£"-J'' (7,14'4)
j=0 3=0

390 SELF-SIMILAR PROCESSES 7.14
where convergence is in the Lp-norm for any 0 < p < a. Consequently,
«P(B)[C(fi)c„] = A(B)en = C(B)[«F(B)c„] a.s. (7.14.5)
Moreover, the left-hand side of (7.14.3) converges absolutely a.s. for a > 1. If
a < 1, the absolute a.s. convergence in (7.14.3) takes place under the additional
assumption that Yl'jLo l^jT < °° f°r some 0 < r < a.
Hint: See Kokoszka and Taqqu (1993c).

Chapter 8
Chentsov random fields
Stochastic processes {X(t), t G En}, n > 1, whose parameter space is the
Euclidean space En, n > 1, are called random fields.1 We shall be interested
in H-sssis fields, that is, in random fields that are self-similar with index H and
have stationary increments in the strong sense. Although the definition of "self-
similarity" in En is analogous to that in E1, there are several possible definitions of
"stationary increments" in En. Here, we suppose that the increments are invariant
under all Euclidean rigid body motions.
In Section 8.1 we introduce the iJ-sssis fields. A well-known example is
the Levy Brownian motion whose autocovariance function is EX(t)X(s) =
k{¥\\ + \\4 - II* - SID- !t was introduced by Paul Levy in 1948 (Levy 1965)
and given a geometric construction by Chentsov (1957). Chentsov's construction
allows the field to be defined as M(Vt)i * £ En, where M is a Gaussian random
measure and Vt is the set of all hyperplanes separating the origin zero from the
point t. We define the Chentsov fields in Section 8.2 by generalizing Chentsov's
construction: we let the measure M be SaS, 0 < a < 2, and consider arbitrary
measurable sets Vt. We then consider the important subclass of H-sssis Chentsov
fields, defined only for H < l/a. A first example, the Levy-Chentsov random
field, is given in Section 8.3. That field is ^-sssis and is the stable analog of the
Levy Brownian motion with the same Vts.
A second example, Takenaka fields, is obtained in Section 8.4 by letting the
Vts be all spheres that separate 0 from t. By choosing the control measure m
judiciously, the Takenaka fields become F-sssis with H < l/a, and are then
called (a, H)-Takenaka. They are the stable counterpart the L£vy fractional
Brownian field, defined for H < 1/2, whose autocovariance function equals
EX(t)X(s) = {{\\t\\™ + ||s|p* - ||t - s||2*}.
'To lighten the notation of the chapter we do not use bold letters to denote elements of En.

392 CHENTSOV RANDOM FIELDS 8.1
In Sections 8.5 and 8.6 we give properties of the finite-dimensional
distributions of, respectively, Chentsov fields and if-sssis Chentsov fields. We show
that the spectral measure is concentrated on a finite number of points and that
normalized SaS iif-sssis Chentsov fields with the same a and H have the same
two-dimensional distributions.
In Section 8.7 we consider the codifference, which provides information on the
two-dimensional distributions. Because the codifference is defined for stationary
stochastic processes, we first consider the H-sssi process {X(ue), u € R},
which is the projection ofan.ff-sssis Chentsov field {X(t), t £ Rn } in a direction
e € Rn, and then consider the corresponding increment process {Y(u) = X((u +
l)e) — X(ue), u £ K}. We show, using the codifference, that any projection
process {X(ue), u S R} obtained from a Chentsov field is different from the
linear fractional stable motion and from other if-sssi processes considered in
Chapter 7.
The last section deals with Takenaka processes on [0, oo). In this case, the
two-dimensional distributions determine all the higher-dimensional distributions.
8.1 Self-similar fields with stationary increments in
the strong sense
The definition of self-similarity for T = Rn is analogous to the one for T = R1.
Definition 8.1.1 A random field {X(t), t e Rn} is self-similar with index H > 0
(tf-ss) if {X(at), t € R"} = {aHX(t), t e Mn} for all a > 0, where, as usual,
= denotes equality of the finite-dimensional distributions.
The extension of the notion of stationary increments to Mn, n > 1, is more
delicate. Recall that a process {X(t), t € R} has stationary increments if
{X{t + s) - X{s), t e R} = {X(t) - X(0), t € R}, for all s € R,
i.e., if the finite-dimensional distributions of the increments are invariant under
translation. Translations are the only Euclidean rigid body motions in R, but in
Rn, the Euclidean rigid body motions include all rotations and translations. Let
<?(Rn) denote the group of Euclidean rigid body motions in Rn.
Definition 8.1.2 The random field {X(t), t € Rn} has stationary increments in
the strong sense (sis) if
{*(<?(*)) - X(g(0)), t e Rn} = {X(t) - X(0), t E R"},
for all Euclidean rigid body motions g e £(R").

8.1
SELF-SIMILAR FIELDS
393
Notation. A random field which is both H-self-similar and has stationary
increments in the strong sense is denoted H-sssis.
Example 8.1.3 Levy fractional Brownian field. LetO < H < 1 and a > 0. The
Gaussian field X = {X(t), t e Rn} with mean 0 and autocovariance function
EX(t)X(s) = y{||i||2H + \\s\\™ - \\t-s\\2H}, t,s€ Rn, (8.1.1)
is called the Levy fractional Brownian field. There are several ways to verify that
X is well defined. One could note, for example, that the right-hand side of (8.1.1)
is non-negative-definite (Lemma 2.10.8) and hence defines an autocovariance
function. One can also construct X as the integral
a0 / (||t - i||"-t - \\x\\H-?)M(dx), (8.1.2)
where M is a Gaussian random measure on Rn with Lebesgue control measure
and (To is a constant proportional to a (Exercise 8.1).
The Levy fractional Brownian field X is an extension of fractional Brownian
motion to M.n. In fact,
Proposition 8.1.4 Let 0 < H < 1. Then the Levy fractional Brownian fields are
the only H-sssis Gaussian fields on Rn.
Proof: Let {X(t), t 6 Rn} be an H-sssis Gaussian field, if-self-similarity
implies that for each a > 0, X(0) = X{a0) = aHX{0) and hence X{0) = 0.
Now any unit vector eeR" can be expressed as e = ge(eo) where eo =
(1,0,... ,0) e M.n and where ge e 5(Rn) is the rotation that maps e0 onto e.
Since
EX(e) = EX(ge(e0)) = E(X(ge(e0)) - X(ge(0))) = EX(eQ),
we obtain
E(X{s + t)- X{s)) = E{X{t) - X{0)) = EX{t) = ||i||HEX(e0). (8.1.3)
By H-self-similarity, on the other hand,
E(X(s +1) - X(s)) = (||s + t\\H - \\sf)EX{eQ). (8.1.4)
Equating (8.1.4) and (8.1.3) proves that EX{t) = 0.

394
CHENTSOV RANDOM FIELDS
8.2
Similarly, EX{t)2 = \\t\\2Ha2 for all t 6 Rn where a2 = EX2(e0).
Therefore for all t,s e R",
E(X(t) - X(s))2 = E(X(t -s)- X(0))2 = EX2(t - s) = \\t - s\\2Ha2
and
EX(t)X{s) = l-{EX2{t) + EX2{s)-E{X(t)-X{s))2}
= a^{\wH+\\s\r-\\t-stHh
which is the autocovariance function of a L£vy fractional Brownian field. I
8.2 Chentsov random fields
We want to study SaS random fields of the form X(t) = JE 1 yt (x)M(dx), t €
Rn, where M is a SaS random measure with control measure m and the Vis are
sets parametrized by t S Rn.
Definition 8.2.1 Let 0 < a < 2, {E, £, m) be a measure space, £ be non-trivial,
M be a SaS random measure with control measure m and {Vi, t € Rn} be a
family of measurable subsets satisfying
m(Vt) < oo for all teW1.
The random field
x{t) = M{vt), tsr,
is called a SaS Chentsov field. The measure m is called the associated measure.
Example 8.2.2 Suppose a = 2 and n = 2 and let M be a Gaussian random
measure on (R+)2 with Lebesgue control measure. The so-called Brownian sheet
is the Chentsov field {X(t) = M(Vt), t € (R+)2} with
Vt = {(xi,x2): 0 < ii < t, 0 < x2 < t}, t e (R+)2.
X has mean zero and autocovariance function EX(s)X(t) = (st A t\) (s2 A t2)
where s = (si, s2), * = (*i, *2). It is self-similar with H = 1 but it does not have
stationary increments in the strong sense.
We will be interested in Chentsov fields that are self-similar and have stationary
increments in the strong sense. Before showing that such fields exist, let us
determine the restrictions that the additional assumption of if-sssis imposes on
the associated measure. Fix 0 < a < 2, H > 0 and let {X(t), t € Mn} be a
SaS, JZ-sssis Chentsov field with associated measure m.

8.2
CHENTSOV RANDOM FIELDS
395
Proposition 8.2.3 H-ss implies
m(Vat) = aaHm{Vt), for all a > 0, te Kn. (8.2.1)
The stationarity of increments in the strong sense implies
m{Vg{t)AVg{0))=m{VtAV0), for all <? € G(Rn), t£Mn, (8.2.2)
where A denotes the symmetric difference.
PROOF: H-ss implies M(Vat) = aHM{Vt) for any a > 0 and any fixed t € Rn.
Equating the characteristic functions yields m(Vat) = aQHm(Vt).
Stationarity of the increments in the strong sense implies
M(Vg{t)) - M{Vm) ± M(Vt) - M(V0) (8.2.3)
for any fixed t and g £ Q(M.n). Letting Vc denote the complement of V, we have
M{Vg{t)) - M(Vg{0))
= M(vg{t) n vfl(0)) + M(Vg{t) n v;(0))
- M(Vg{0)f)VgW) - M(VgiQ)nVgc{t))
= M(Vm n V^oj) - M(V9(0) n y;(t)), (8.2.4)
Since these last two terms are independent, we obtain
-\nESxP{i(M(Vg{t))-M(Vg{0)))}
= m(Vg{t) n Vgc(0)) + m(Vg{0) n Vgc{t))
= m((v9(t)nvgc(0))u(vfl(0)nv;(t)))
= m(Vg{t)AVg{0)). (8.2.5)
Relation (8.2.3) establishes (8.2.2). I
Remark. Setting t = 0 in (8.2.1) gives
m(V0) = 0,
i.e., X(0) = M(V0) = 0 a.s.. Then (8.2.2) becomes
m(VgW A Vg{0)) = m{Vt), Vg € 0(Rn), t e Rn.
The measure m and the family {Vt, iGR"} induce apseudo metric
d{t,s) = m(VtAVs)

396 CHENTSOV RANDOM FIELDS 8.2
in Rn. Indeed, d(t, t) = 0, d(t, s) = d(s, t) and d(f, s) < d(t, u) + d{u, s). To
establish the last relation, note that
VtAV, = (Vt A Vu) A (VL A V.)
c (K,Ayu)u(KAi/s).
The function d does not define a metric because d(t, s) = 0 does not imply t — s
in general. If X is a if-sssis process, however, then d is a metric. Indeed,
Proposition 8.2.4 H-sssis implies
d(t,s) = m(VtAVs) = c\\t-s\\aH,
where c = d(eo, 0) and e<> = (1,0,..., 0).
PROOF: Relation (8.2.2) implies d(g{v), g(0)) = d(v,0) for any Euclidean rigid
body motion g in K". Setting g(v) = v — u gives d(v — u, —u) = d(v,Q) and
setting t = v — u, s = -u yields
d(t,s) = d(t-s,Q). (8.2.6)
Now t — s = ||;t — s||e where e is the unit vector in the direction t — s. However,
relation (8.2.1) implies
d(t -s,0)= d(\\t - s\\e,0) = ||i - s\\aH d(e,0). (8.2.7)
Now let g be the rotation around the origin that maps the unit vector e into the
unit vector eo = (1,0,... ,0). Since g(0) = 0, we obtain by using (8.2.1),
d(e,0) = m(VreAVro)=m(Vfl(e)AV9(0))
= m(VeoAV0) = d(e0,0). (8.2.8)
Relations (8.2.6), (8.2.7) and (8.2.8) prove the proposition. ■
Corollary 8.2.5 lf{X{t), t € Rn} is an H-sssis SaS Chentsov field, then
H< 1/q.
PROOF: Proposition 8.2.4 and the triangle inequality imply c\2t\aH = d(2t,0) <
d(2t, t) + d(t, 0) = 2c\t\ali, i.e., 2aii < 2 or H < l/a. I
Corollary 8.2.5 implies that the Levy fractional Brownian field cannot be
represented as a Chentsov field when 1/2 < H < 1. We will see that it can be so
represented when 0 < H < 1/2 (Examples 8.3.3 and 8.4.5).

8.2
CHENTSOV RANDOM FIELDS
397
The Levy-Chentsov and (a, if )-Takenaka fields defined below provide
examples of SaS if-sssis Chentsov fields with H = l/a and 0 < H < 1/a,
respectively.
Propositions 8.2.3 and 8.2.4 were established using only the marginal
distributions of the field X. They provide necessary conditions for H-sssis. The next
result gives sufficient conditions for if-sssis and involves the finite-dimensional
distributions of X.
The following notation will now be used:
For d > 1, consider the set of all functions
Zd = { A : {1,2,... ,d} - {0, l}d\(0,0,... ,0)} (8.2.9)
and let 5 — (5(1), 5(2),... ,6{d)). Each component of 5 € Id is either 0 or l,but
the components are not all zero. Moreover, if B is a set, let
[ B° = Bc = the complement of B.
Thus, {Bc)' = B° and (Bc)° = Bx = B.
Theorem 8.2.6 Set 0 < a < 2, 0 < H < l/a. Let (E,£,m) be a measure
space, M be a SaS random measure on (E, £) with control measure m and
{Vt, t £ M.n} be a family of m-measurable sets. Then
X{t) = M(Vt), t£R",
is an H-sssis SaS Chentsov process if the following three conditions hold:
(0 m(Vt) < co for any t € En;
O'O
™(f)V$?)=*aHrn(rivt!k)) (8-2-10)
foranytut2,...,td£M.n, 5 € Sd, d> 1, a > 0;
070
™(n(c))A^))=m(n^(fc))' (8-2J1)
foranytu...,td 6Rn, 5 € Ed, d> 1 and for any g € S(Rn).

398 CHENTSOV RANDOM FIELDS 8.2
Proof: Condition (i) ensures that {X(t), t e Rn} is a SaS Chentsov field. To
prove that it is H-ss, let t\,..., td G Rn and notice that each set Vt,,..., Vtd can
be expressed as a union of disjoint sets as follows:
*,- u h<k)>
where the union is over all functions 6 € AQ such that 6(j) = 1 (see Exercise
8.2).
We shall now prove that the characteristic function of the vector
(X(aU),...,X(atd)), a > 0, equals that of (aHX(ti),. ..,aHX(td)). We
have
d
EexpU^QjX-iatj)}
i=i
d
= Ecxp{iJ20jM(Vatj)}
= E^{i±ej £ m((\v™)}
= n^-pKEw))^(n^))}
d
= ••• = £;exp|i^^aHX(i:,)},
by retracing the preceding steps.
We establish the stationarity of the increments by showing that for any
g e £(Mn), the characteristic function of (X(g(t\)) - X(g(0)),..., X(g(td) -

8.2
CHENTSOV RANDOM FIELDS
399
X{g(0))) equals that of (X(ti),.. .,X{td)). (Notice that (ii) implies m(V0) = 0.)
We have
d
Ew{iY,ei(X{9{ti))-X{g{0)))}
3 = 1
d
= E^v{iJ2S3(M(Vg{tj))-M(Vg{0)))}
i=i
= Eexpji f^ 03iM{VgM n V;(0)) - M(V9c(tj) n Vg{0)))J as in (8.2.4)
= w^jV u nte,nvs0)))
11 j=\ V Vfi:6(j) = ] fc=l V / '
M u n((^))'(fc)nv,(0,)))
= n«p{-i e Wnte^
xexP{-| J2 93\Qm(f]((vg\tk)) nVg{0))) las in (8.2.12)
= II «p{-| J>*(j)|am(n V/S) AV^(0)) j ^in(8.2.5)
= IIexp{-lEfl^')rm(nvtt(fc))}by(iii) (8.2.13)
d
= ■•• = £?exp|i53fljX(ij)}, (8.2.14)
after retracing our steps. This concludes the proof. I
The preceding proof also establishes:
Corollary 8.2.7 Let {X(t), t e Kn} be a Chentsov field with associated measure
m. The characteristic function of the vector (X(t\),..., X(td)), d > 1, eaua/.s
3 = 1 «6Sd •• j:«(j)=l Vfc=l /J
(8.2.15)
and w therefore determined by the numbers m(f]k=i Vt^ '), S € Sj.

400
CHENTSOV RANDOM FIELDS
8.3
8.3 Example: the Levy-Chentsov random field
As a first example, we construct the L6vy-Chentsov random field, which is an
extension to a < 2 of the LeVy fractional Brownian field defined in Example
8.1.3.
Let Sn denote as usual the ((n — 1)-dimensional) unit sphere in Kn and let s
be a point on Sn. A hyperplane h in M" which does not contain the origin is a set
of dimension n — 1 parameterized by (s,r) € Sn x R+ where E+ = {r: r > 0}.
Figure 8.1: The hyperplane h is denned
by the parameters s and r.
Here, it will be more convenient to regard the hyperplane h not as the set {x €
Mn: < x, s >= r}, where < , > denotes the scalar product in ]Rn, but as the pair
of parameters (s, r) s Sn x M+. We are now in position to define the measure
space (E, £, m) and the sets Vt, t € M". Let
E = all hyperplanes h in Kn of dimension n — 1
that do not contain the origin
= {(s,r): s 6 S„, 0 <r < oo}
— Sn x R+,
6 = Borel u-field on E,
and let m be the measure on (E, S) given by
m(ds, dr) = ds dr
where ds denotes the Lebesgue (i.e., uniform) measure on on (m(Sn) =
area of Sn). The measure m is a-finite since
oo
{s e Sn, r > 0} = \J {s e Sn, i < r < i + I}.
t=0

8.3
EXAMPLE: THE LEVY-CHENTSOV RANDOM FIELD
401
If B is a set of hyperplanes in £, then
m{B) = / dsdr. (8.3.1)
Jb
Hence:
(i) m({h}) = 0 (the measure of a single hyperplane is 0).
(ii) A set of hyperplanes, all containing a given point has m-measure 0.
(iii) A set of parallel hyperplanes has m-measure 0.
A rigid body motion g £ Q(M.n) induces naturally a map g' on Sn x R+. Since
m(g'(B)) = m(B), (8.3.2)
we say, simply, that the measure m is invariant with respect to Euclidean rigid
body motions of hyperplanes. Thus, in particular, the measure of all hyperplanes
that intersect segments of equal length is the same.
Finally, for each tel", let
Vt = all hyperplanes h separating the origin 0 and the point t
= {{s,r): 0 <r < < s,t >}
and V0 = 0. Observe that
m{Vt) = / Lebi(r: 0 <r « s,t >)ds
= ]- [ \<s,t>\ds = c\\t\\, (8.3.3)
1 Js„
wherec= (1/2) JSn < s,e0 > ds and e0 = (1,0,...,0) G Sn.
Definition 8.3.1 A Levy-Chentsov field {X{t) = M{Vt), t e Mn}isaSa5', 0 <
a < 2, Chentsov random field with the associated measure m and the sets {Vt, t €
Kn} defined above.
Theorem 8.3.2 A SaS, 0 < a < 2, Levy-Chentsov field {X{t) = M(Vt), t G
Rn} has the following properties:
• It is --sssis.
a
• There is a constant c> 0 such that
d{t,s) = m{Vt&V0)=c\\t-s\\.
• ]fn=\, non-overlapping increments are independent.

402
CHENTSOV RANDOM FIELDS
8.4
PROOF: Conditions (i), (ii) and (iii) of Theorem 8.2.6 hold with H = 1/a. (The
verification of condition (ii) is similar to that of (8.3.3), and condition (iii) holds
because m(Vo) = 0 and m is invariant under rigid body motions of hyperplanes.)
Hence X is an ^-sssis random field. The relation d(t, s) = c\\t — s\\ follows from
Proposition 8.2.4. Finally, set n = 1 and let 0 < t\ < i2 < *3 < U- Hyperplanes
that pass through an interval can be identified here with that interval, and since
the intervals [t\, t2] and [t3, £4] are disjoint, we conclude from Theorem 3.5.3 that
X(t2) - X(t\) is independent of X(t2) - X(tx). ■
Example 8.3.3 When a = 2, the L£vy-Chentsov field {X{t) = M(Vt), i€Rn}
is Gaussian with mean 0 and autocovariance
EX{t)X{s) = y {p|| + |M| - ||t - s||}, t, s e R",
by Proposition 8.1.4. It is the Levy fractional Brownian field with H = 1/2, and is
usually known as Levy Brownian motion. If, in addition, n = 1, then EX(t)X{s)
equals either min(£, s), if i and s have the same signs, and equals 0 if they have
opposite signs. Hence, {X(t), t € R1} is Brownian motion.
8.4 Example: Takenaka random fields
Although, in the preceding section, we defined Vt as the set of hyperplanes
separating 0 and t, we shall now take Vt to be the set oi hyper spheres that separate
0 and t. Our goal is to construct a family of SaS Chentsov fields that are iJ-sssis
with H < 1/a.
A (hyper) sphere in Rn of dimension n — 1 can be represented by a pair (x, A)
where x € Rn is its center and A € R+ is its radius. It will be more convenient
here to regard the sphere not as the subset {y: \\y - x|| = A} of Rn but as the pair
of parameters (x, A) 6 1" x 1+.
Define
E = all spheres in E" of dimension n — 1
= {(x,A):x€Rn, 0<A<oo} (8.4.1)
= Rn x K+,
£ = Borel a-field on E,
and for each t € Rn, let
Vt = all spheres in E separating the origin 0 and the point t
= {(x,A): ||x|| <A}A{(x,A): ||x - t|| < A}. (8.4.2)

8.4
EXAMPLE: TAKENAKA RANDOM FIELDS
403
The set Vt is illustrated in Figure 8.2 in the case n = 1.
X
•yy w
0 t
Figure 8.2: The set Vt for t € R1.
To understand the analytic definition of Vt, note that the set Vt excludes spheres
that contain both 0 and t or that contain neither 0 nor t. Thus, for a sphere (x, A)
to be in Vj, we must have either
||0-x|| < A and \\t - x\\ > A
(0 is in the sphere but t is not) or
\\t-x\\ < A and ||0'- x|| > A
(t is in the sphere but 0 is not).
Definition 8.4.1 Let 0 < a < 2. If {E,S) and {Vt, t € Md} are defined
as in (8.4.1) and (8.4.2), respectively, then the SaS Chentsov field {X{t) =
M(Vt), t 6 M.n} is called a (SaS) Takenakafield.
A Takenaka field can have any associated measure m, as long as m(Vt) <
oo, t e R". Because we are interested in self-similar fields, we shall consider the
family of associated measures m = mp, 0 < 0 < \, where
mp{dx,d\) = A^-"-'dx dX. (8.4.3)
Lemma 8.4.2 (a) Fix 0 < 0 < 1. Then m0(Vt) < oo for all t € W1 and,
moreover,
m0(yt) = -^ / (Nrn - II* - tf-n)dx (8-4-4)
which equals
n - 0 A" v 2 fc=2 fc=2
(8.4.5)
(fc) Ifmp(Vt) < oofor some t ^ 0 and some -oo < /3 < oo, f/ien 0 < 0 < 1.

404 CHENTSOV RANDOM FIELDS 8.4
PROOF: Set lip{d\) - X0 " ldX so that mp(dx, d\) = np(d\)dx and define,
for each x e Rn, Vtlx) = {A: A > ||x - *||} A {A: A > ||x||} C R1. Then
mp(Vt) equals
/
Hp(Vtlx))dx
R"
= / ^(Vt{x))dx+ [ MVt{x))dx
= 2 / H(l(Vtlx))dx
./{x€R":||x||<||x-t||}
r / rU1-^ \
= 2 ( X0-n-ldX)dx
./{x€R":||x||<||x-t||}\/||x|| '
= ZTTi[ ' (\\x\f-n-\\x-t\f-n)dx,
rc-P./{x€R":||x||<i|x-t||}
establishing (8.4.4).
Notice that the integrand in (8.4.4) diverges as x —» 0 or x —> oo. As x —> 0,
the integrand behaves like ||o;||^-n, so the integral converges in a neighborhood
of 0 if and only if 0 - n + (n - 1) + 1 > 0, i.e., 0 > 0. As x —> oo,
the integrand behaves like ||x||^-n_1, so convergence takes place if and only if
0 - n - 1 + (n - 1) + 1 < 0, i.e., 0 < 1.
Finally, we prove (8.4.5). Let < , > denote the scalar product in Rn and notice
that the inequality ||x|| < ||t — x\\ is equivalent to < x, x > < < x — t, x — t >
= < x,x > -2 < t,x > + < t,t >, that is, to < t,x > < \\t\\2/2. Let
t = \\t\\e where e is a unit vector in R".
If t ^ 0, the change of variable x —» x/\t\ in (8.4.4) yields
mp{Vt) =Wt [ (Hxll"-" - ||x - ef~n)dx.
n- p 7{x€R":<e,x> < 1/2}
Since Lebesgue measure in Mn is invariant under rotations, the integral does not
depend one. Replacing e by (1,0,.. .,0), we see that m^(Vt) equals
-^h OC1*) -((x.-l^ + ^Jxl) \dx.
n-/3 7{x€R":x,<l/2}LVfct ' k^l
The change of variable z\=x\ — \, zk = xfc, k > 2, yields (8.4.5). I
When n = 1, we have (see Exercise 8.3)
m0{Vt) = W=T)22'"ltlP- (8A6)

8.5
PROPERTIES OF CHENTSOV RANDOM FIELDS
405
Definition 8.4.3 Let 0 < a < 2 and 0 < /3 < 1. The SaS Takenaka field
{X(t) = M(Vt), t <E Rn} with associated measure mp is called an (a,H)-
Takenaka field, where H — 0/a € (0,1/a).
Theorem 8.4.4 An (a, H)-Takenaka field, with 0<a<2and0 < H < 1/a, is
H-sssis.
Proof: We must verify conditions (i), (ii) and (iii) of Theorem 8.2.6. Condition
(i) follows from Lemma 8.4.2. To verify condition (ii), one could return to the
proof of Lemma 8.4.2, but it is easier to note that
Vat = (Or, A): ||x|| < A} A {(a;, A): ||a; - at|| < A}
= {(ax, aX): \\x\\ < A} A {(ax, aA): ||x - t\\ < A},
and thus Vat = aVt, where aVt means that every element of Vt is multiplied by a.
Similarly, V£ = aVtc and Va = aV, where Va = f]dk=l V^\
Therefore,
m0{Va)= f \p-n-ldxdX = a0m0{Vl)
Jva
since d(ax) = andx and d(aA) = adA.
We now verify Condition (iii) of Theorem 8.2.6. Since the transformation
(x, A) —> (g~l (x), A) is an isometry, its Jacobian equals 1 and therefore
^ (n (c> a ^(o)))=^ (n (<{k) * *))=m„ (n ^(fc))
using Vo = 0.
The three conditions of Theorem 8.2.6 hold with H = /3/a and, therefore, the
SaS Takenaka field {X{t) = M(Vt), t € Kn} with associated measure mp is
/3/a-sssis. I
Example 8.4.5 A (2, H)-Takenaka field is a Levy fractional Brownian field
(Proposition 8.1.4).
8.5 Properties of Chentsov random fields
In this section we investigate the properties of the finite-dimensional distributions
of Chentsov fields on R". In Theorem 8.5.1 we show that the spectral measure is
discrete and in Theorem 8.5.3 we investigate what happens when a varies.

406
CHENTSOV RANDOM FIELDS
8.5
Theorem 8.5.1 Fix 0 < a < 2 and let {X(t), t € Rn} be a SaS Chentsov
random field with associated measure m. Then the spectral measure V of the vector
(X(t\),..., X(td)), d > 1, is discrete and concentrated on at most 2(2d — 1)
points of the unit sphere Sd- These points are iC5. where
£« = _! (6(1),..., 5(d)), 6el.d,
(EJ=i*(j)),/2
and where 2<i is defined in (8.2.9). The point masses of the measure F are
n-cs) = rxc*) = 5(E*(j))"/2™(n <{k))- (8-51)
PROOF: The characteristic function.of (X(t\),..., X(td)) is
a
Etxp^Y^OjXitj)}
j=i
= exp{-El E 0i\a™(r\V?klk))} (as in (8.2.12))
*• field j:fi(i)=i V=i ' -*
by Corollary 8.2.7. The theorem follows by identifying this expression with
expj- J \0lSl+--- + edSd\a\r{ds). I
The distribution of a SaS Chentsov field {X(t) = M(Vt), t € En} clearly
depends not only on the control measure m of M but also on the value of a.
To make the dependence on a explicit, we now write Xa(t) = Ma(t), teR",
denoting by Ma a SaS random measure with control measure m. What happens
when we fix m and let a vary?
Definition 8.5.2 Fix (E, S, m) and a family of sets {Vi, t€ Kn}. The family of
random fields
{{Xa(t) = Ma(Vt), t e E"}, 0 < a < 2}
is called a conjugate class of Chentsov fields. The field {X2(t), t € M.n} is the
Gaussian element of the class.
Theorem 8.5.3 Let {{Xa(t), t 6 Rn}, 0 < a < 2} and {{X'a(t), t €
M"}, 0 < a < 2} be two conjugate classes defined by ((E,£,m), {Vt}) and

8.6 PROPERTIES OF tf-SSSIS CHENTSOV RANDOM FIELDS 407
{(£", £', m'), {V/}}, respectively. If the Gaussian elements X2 and X'2 are
identically distributed, then for any 0 < a < 2, the fields Xa and X'a have the same
two-dimensional distributions.
Moreover, if{Xao(t), t <E Rn} = {X'ao(t), t e Kn} for some 0 < a0 < 2,
then {Xa{t), t e R"} = {^(t), * 6 Mn}/ora//0 < a < 2.
Proof: Xa has two-dimensional characteristic function
Eexp{e1XQ(h)+62Xa(t2)}
= exp{-{|flI|°m(Vl1 n V$ + \e2\am(Vt* fl Vh) + |0, + 62\am{Vu n Vh)}}.
(8.5.2)
The Gaussian element has characteristic function
E^p{elx2(to+e2x2(t2)} = Sxp{-{l-eWtl+0i92atl,t2+^cr2t2}}, (8.5.3)
where a\ = -E(X2(t))2 and oUM = EX{ti)X{t2). Identifying (8.5.2) with
q = 2 and (8.5.3), we obtain
Tn(VtinVQ+m(VtinVt2) = \o\,
m(^nv(!) + m(yt|n^) = \a\v
2m(VtinVt2) = atlttl,
( m(VtinVQ = \(al-otlM),
< ^nV(2) = }K-at|,(!), (8.5.4)
w m(Viinl,t1) = }ff|l,v
Any two conjugate families with equally distributed Gaussian elements have the
same as and, hence, the same values for m(Vtl D V£), m(Vtct C\ Vt,), m(Vt, r\Vt2).
By virtue of (8.5.2), they have the same two-dimensional characteristic functions.
The last part of the theorem follows from Theorem 8.5.1. I
8.6 Properties of iiT-sssis Chentsov random fields
Whereas the preceding section covered arbitrary SaS Chentsov fields, we consider
in this section SaS Chentsov fields that are also H-sssis, for example, (a, H)-
Takenaka fields.

408 CHENTSOV RANDOM FIELDS 8.6
Proposition 8.6.1 Fix'oto G (0,2] and H0 G (0, l/a0] and let {X(t), t G Rn} be
a SaoS, Ho-sssis Chentsov random field. Then the conjugate class {{Xa(t), t G
Kn}, 0 < a < 2} with Xao = X is well defined and, moreover, each member
Xa, 0 < a < 2, is a SaS, Ha-sssis Chentsov random field, with Ha = aoHo/a.
PROOF: Fix 0 < a < 2 and consider {Xa(t) = Ma(Vt), t e Rn} where Ma
has control measure m. Since aHa = ao-ffo, the control measure m satisfies
the conditions of Theorem 8.2.6 and therefore Xa is a SaS #a-sssis Chentsov
random field. I
Normalized SaS random fields have identical one-dimensional distributions.
The next result states that if they are if-sssis Chentsov with the same H, then they
also have identical two-dimensional distributions.
Theorem 8.6.2 Fix 0 < a < 2 and 0 < H < \/a. Then any two SaS,
H-sssis Chentsov random fields have, up to a multiplicative constant, the same
two-dimensional distributions.
PROOF: To avoid confusion with previous notation, let ao and Ho denote,
respectively, the a and Fin the statement of the theorem. Let{A"(i) = M(Vt), t G E"}
and {X'(t) = M'(V/), t e Kn} be the two random fields with associated
measures m and m', respectively. Consider, as in Proposition 8.6.1, the conjugate
class {{Xa(t), t G R"}, 0 < a < 2} with Xao = X, and also the conjugate
class {{X'Q(t), t G Rn}, 0 < a < 2} with X'ao = X'. Proposition 8.6.1 states
that the Gaussian elements X2 and X'2 are both i^-sssis with H% = aoH0/2 and
hence they are both L6vy fractional Brownian fields. Since the L6vy fractional
Brownian fields with same self-similarity index differ only by a multiplicative
constant C, we have X2 = CX^. Therefore, by Theorem 8.5.3, the relation
(A-Q(t,),XQ(i2)) = (CXa(ti), CXa(t2)) holds for all 0 < a < 2 and, in
particular, for a = ao. B
What happens if we restrict the argument t G Kn in (a, H)-Takenaka or L6vy-
Chentsov fields to a linear subspace Rm C Rn with m < n? Do we obtain a new
type of random field? The answer is in the negative.
Theorem 8.6.3 Fix 0 < a < 2, 0 < /? < 1, m < n, set H = P/a, and
let {Xn(t), i G Rn} denote an (a, H)-Takenaka field with associated measure
dm%(xn, A) = A/3_n"-1cixnc(A, xn G Rn. Then the random field
Ym(t) = Xm(t,0,...,0), t&Rm,

8.6 PROPERTIES OF tf-SSSIS CHENTSOV RANDOM FIELDS 409
has the same finite-dimensional distributions as {CXm(t), t G Em}, where
C is a constant and Xm is an (a, H)-Takenaka field with associated measure
dm^(xm,X) = Xp-m~ldxmdX, xm e Rm.
A similar statement is true if Xn and Xm are Levy-Chentsov fields.
PROOF: We shall consider Takenaka fields (the argument is similar for a Levy-
Chentsov field). We may assume m = n — 1 (the general result follows by
induction). Let xn = {xm,x\) G W1, where xm € Rm, x\ € R1 and let
i = (t, 0) G K", where t G Km. Then Ym(t) = Mn{Vtn), t G Rm, with
Vtn = {(xm,xuX): \\{xm,Xl)\\<\}
A{(xm,a:i,A): ||(xm,a;i) - (i,0)|| < A}
= {(xm,xl,Xy.\\xm\\2<X2-x]}
A{{xm,xuX): ||a;m-t||2 < A2 - x2}
= {{xmAv)-\\xm\\<V, \9\<n/2}
A {{xm, 9, v): \\xm - t\\ < r,, \9\ < tt/2},
where we set rj = (A2 — x2)'^2, sin# = X\/X and, thus, cos# = 77/A. Therefore
V^V^x [-tt/2,tt/2].
Since the Jacobian of the transformation (xx, A) i-> (9, 77) equals 77/ cos 9, we have
rr${V?) = I \vt"{xm,xuX)Xl3-n-ldxmdxldX
= / / \v">{xm,ri)[ - -dxmdr)d9
./-7r/2./R">xR+ ' \COs0j COS9
= / lv1™(im,r;)^-r,l-1da:TndT, f {co%6)m+^d6
iKmxKx J-tt/2
/RmxK+ J -tt/2
= Cm?{Vn,
where C = J^2Jcos9)m+^d9.
t/2
But the finite-dimensional distributions of Xn are determined by the measure
of sets of the form f\dk=i{Vtnk)^k) (see (8.2.12)). Since T transforms these sets
into nfc=i(^iT)6(fc) x [~f > f ]•tne Preceding argument also shows that
Q(V»)«<fc>)=Cm?((Vi?)4(fc))-
ifc=n
d N / d
The result now follows from Theorem 8.5.1. I

410
CHENTSOV RANDOM FIELDS
8.7
8.7 Codifference induced by (a, #)-Takenaka fields
We now turn to the codifference r(-) and to the function I{6\,6i; ■) defined in
(4.7.1) and (4.7.5), respectively. These functions provide information on the
two-dimensional distributions. Since they are defined for stationaiy stochastic
processes only and not for non-stationary random fields, we want to apply them
to stationary stochastic processes induced by an (a, #)-Takenaka random field
{X(t), t e Rn}.
Choose an arbitrary unit vector e € Rn centered at the origin and consider
the projection process {X(ue), ueR1} which is defined on the line spanned
by the vector e. Since the random field X is H-sssis, the projection process
{X(ue), u 6 K1} is if-sssi. Therefore, the increment process
Y(u) = X{{u + l)e) - X(ue), u e E1
is stationary.
The following analysis centers on Y. Our goal is to derive the asymptotic
behavior of the function
i(eue2;u)
= -lnSe^'^+^o)) + lnEeie,Y(u) + lnEemY(o)
= ||0,r(«) + e2Y(o)\\aa - II W«)HS - H^(o)||S (8.7.1)
as u —» 00 and hence that of the codifference
t(u) = -/(1,-1;u)
when X is an (a, i?)-Takenaka field. To simplify the notation, we let m denote
its associated measure (m is the mp of (8.4.3)). The normalization constant
\\Y{\)\\% = ll^((« + m - *(t"0IIS = HM(Ve)IIS = m(Ve)
which appears in the results does not depend on the chosen direction e. Its
numerical value is given by (8.4.5) (with (3 = aH and \\t\\P = 1).
Theorem 8.7.1 Let {X(t), t € Kn} be an (a, H)-Takenaka field, 0 < a <
2, 0 < H < I/a with associated m, let e £ Kn be an arbitrary unit vector
centered at the origin and let I and r refer to the increment process
Y(u) = X((u+l)e)-X{ue), u 6 R1.
Then
J(0,,02;u) = m{Ve)FQ(6u02){\u+l\aH - 2\u\ali + \u- \\aH}, u > 1,

8.7 CODIFFERENCE INDUCED BY {a, H)-TAKENAKA FIELDS 411
where
Fa(9u92) = 10,1° + |02|Q - \9i - 92\a. (8.7.2)
//, in addition, 9\,92 ^ 0, then, as u —> oo,
I{9u92;u) ~ ~m(Ve)Fa(9l,e2)aH{l~ aHK"H~2
and
r(u) ~ -m(VE)(2a-2)aH{1~aH)uaH-2.
PROOF: Suppose, without loss of generality, that 9\ and 92 are not^qual to zero
and assume firstly that n = 1. Since uje process Y is SaS, it is sufficient to take
e = 1. Since Y{u) = M(VU+1) - M(VU) and V0 = 0, we have
H0,y(«) + 02y(o)||s = ||fliA/(K+i) - 0iM(K) + ^M(y,)lla
= |^|Q(m(KAK+i)-m(5„uK))
+ |^r(m(l/)-m(BuuB;))
+ |0, + ^2|Qm(S;) + |0, - 02|Qm(lU
where Bu = (VuXK+i) n V, and B'u = (Vu+i\K) n V\. By stationarity,
(Vu A Vu+i) = m(Vu A Vo) = tti(Vi). Since Lemma 8.7.2 below shows that for
u > 1, B'u = 0, we obtain
HWiO+WOHS = |flir(m(V,)-m(Bu))
+ |02|Q(m(Vi) - m(Bu)) + |0, - ^|Qm(Su), « > 1,
and hence using (8.7.1), we obtain
I*(9i,92;u) = -FQ(0,,02)m(Bu), u > 1, (8.7.3)
where Ia and Fa are given by (8.7.1) and (8.7.2), respectively. (We have written
Ia instead of I to avoid confusion.) We shall evaluate m(Bu) by considering the
analog of (8.7.3) for a = 2.
The field X is a member of a conjugate class that includes a Gaussian element
X2. By Proposition 8.6.1, X2 is 7J2-sssis with H2 = aH/2. The induced
projection process {X2{ue) ,u6l'}is #2-sssi and, hence, is Brownian motion if
H = l/Q(i.e.,ifiJ2 = 1/2) and fractional Brownian motion (see Definition 7.2.2)
ifH< l/a(i.e.,ifif2< 1/2). Let {Y2(u) = X2((u+l)e)-X2(ue), u e K1}
denote its increment process. Letting I2{9\,92;u) be defined as in (8.7.1) (with
a = 2 and Y = y2), we obtain
/2(0i, 02;u) = i{Var(0,y2(u) + 02y2(O)) - Var <0, Y2(u)) - Var (02Y2(O))}
= 0,02£Y2(u)Y2(O).

412 CHENTSOV RANDOM FIELDS 8.7
The process {Y2(u), u€l'}is fractional Gaussian noise,2 with variance
al = 2||y2(l)||2 = 2||M2(ye)||2 = 2m(Ve).
On the other hand, by (8.7.3), we have, for u > 1,
h{ex,62;u) = -F2{eue2)m{Bu), u>\.
Therefore, foru > 1,
Ueue2;u) = E^Mh{eue2,u)
= \Fa{6u82)EY2{u)Y2(0). (8.7.4)
It remains to evaluate EY2(u)Y2(0). Since 2H2 = aH, this covariance equals
EY2(u)Y2{0) = m(V,){|« + l|Q" - 2|u|a// + |« - 1|Q"}
by Proposition 7.2.9 and, as u —» oo,
£y2(u)y(0) ~ m(Vi)aJET(afl" - l)uQ//-2,
as in Proposition 7.2.10. Substituting these expressions in (8.7.4) concludes the
proof in the case n = 1.
Suppose, now, n > 1. Let {Xx(u) = M(VU), u e M.} he. the (a,H)-
Takenaka process on R with associated measure m1. The theorem, as we have
shown, holds for X\ with m(Ve) equal to m'(Vi). If e0 = (1,0,... ,0) € K",
then by Theorem 8.6.3,
{X(ueo), u € M} = {CX'(u), u€R}
in the sense of equality of the finite-dimensional distributions. To determine the
constant C, notice that
m(K„) = ||*(eo)||2 = \C\a\\Xl(u)\\Z = \C\am\Vx),
i.e.,
\cr =
m(Veo)
m>(V,)"
2In Section 7.2.3, fractional Gaussian noise was defined for u = ..., —1,0,1,..., but its
definition can be extended naturally to all of u 6 K'.

8.7 CODIFFERENCE INDUCED BY (a, H)-TAKENAKA FIELDS 413
Therefore the theorem holds for the field X when e = en. But, because of rotation
invariance, for any vector eel™ centered at the origin, we have
{X(ue), «£!} = {X{ue0), u € R},
in the sense of finite-dimensional distributions. Hence our theorem holds for the
field X and arbitrary e. I
The proof used the following:
Lemma 8.7.2 Foru>\, (Vu+l\Vu) n V, = 0.
PROOF: Recall that Vu = Cu A C0 and Cu = {(A, x):\x-u\< A}. Write
(K+AK) n v, = (K+, n v^) n v,
= [(Cu+1 a Co) n (Cu a c0)c] n [C, a c0]
= [{(Cu+1nc0c)u(c=+1nCo)}n{(c«uCo)n(C„uc0c)}]
n [(C, n c0c) u (Cf n c0)]
= [{(Cu+1 n C0C) u (cs+1 n c0)} n {(c° n c0c) u (Cu n c0)}]
n [(d n c0c) u (Cf n c0)] .
= [(Cu+, n c0c n c^) u (C£+, n c0 n c„)]
n [(C,nc0c)u(c?nc0)]
= (C0C n c, n c^ n cu+1) u (C0 n cf n c„ n c^+1).
Now (A, x) G Cq n Ci n C^ n Cu+i iff simultaneously
|i| > A, \x - 1| < A, |i - u| > A, |x - u - 1| < A. (8.7.5)
Notice that inequalities (8.7.5) hold iff
A<x<A+l andA + u<a;<A-|-l-l-u. (8.7.6)
If u > 1, there are no A and x satisfying (8.7.6), so C£ n C\ n C^ n Cu+i = 0.
In a similar way, one can show that Co n C° n Cu n Cu+i =0. I
We have evaluated the asymptotic behavior of the codifference of a number
of increment processes and found the following. For the projection processes of
(a,ff)-Takenaka random fields, the codifference behaves like uaH~2. For the
linear fractional stable noise, it behaves either like uaH~a or like ■uh_^i/q)_1
(see Theorem 7.10.1 and a remark following the theorem). For the log-fractional
noise, where/f = 1/q, it behaves like ul~a (Theorem 7.10.2). It does not tend to
zero for either the real harmonizable or the sub-Gaussian fractional stable noise
(Propositions 6.7.1 and 4.7.4 respectively). Therefore we have:

414 CHENTSOV RANDOM FIELDS 8.8
Corollary 8.7.3 Projection processes of (a, H)-Takenaka random fields, 0 <
a < 2, 0 < H < 1/q, are different from the linear fractional stable motion, from
real harmonizable fractional stable motion and from the sub-Gaussian fractional
stable motion.
8.8 Takenaka processes on [0, oo)
We shall now consider Takenaka processes on [0, oo), i.e., Chentsov fields
{X(t) = M(Vt), t > 0} where the Vts are defined in (8.4.2). We have seen
that any d-dimensional distribution of a SaS Chentsov field on Rn has a spectral
measure concentrated on at most 2(2d — 1) points of the unit sphere Sd- One can
say more if that field is a Takenaka process on [0, oo).
Lemma 8.8.1 Let 0 < a < 2. If{X(t) = M(Vt), t > 0} is a SaS Takenaka
process, then the spectral measure F of {X(t\), ...,X(td)) is concentrated on at
most d(d + I) points of Sd-
PROOF: The spectral measure of a Chentsov process given in (8.5.1) involves
the sets (\=i Kk i <5 € Zd. It is easy to see that for any 0 < t\ < t2 < £3 < 00,
Vt,nV£nVtj=0 (8.8.1)
and hence it suffices to consider sets of the form
*p,*=(n^)n(fW)n( n ytX i<p<i<d. (8.8.2)
These sets are disjoint (Rp,q D Rp><g>) — 0, if (p, q) # (p',q'), and there are
d(d+ l)/2 of them. The (symmetric) spectral measure T is therefore concentrated
on at most d(d + 1) points. I
Remark. The proof fails if the process X is defined on E1 instead of E+ because
Relation (8.8.1) does not hold, for example, when t\ < t2 < 0 < £3. However,
Vt, n Vtc2 n Vt, n V£ = 0 and V£ n Vu n V£ n Vu = 0 for any t, < t2 < t3 < t4,
and so the measure is concentrated on at most 2d(d — 1) + 2 points in this case.
Let 0<a<2, 0</3< 1, H = 0/a. The proof of the preceding lemma
suggests how to construct SaS, H = /3/a-sssis Chentsov processes which are
not (a, #)-Takenaka.

8.8
TAKENAKA PROCESSES ON [0, oo)
415
Example 8.8.2 Let n = 1 and define a SaS Chentsov process {X(t) =
M(Vt), t>0} with
Vt = CtA C0,
where
Ct = {{x,X) 6 1' xM+: -A<x-t< -A/3 or A/3 <x-t< A} (8.8.3)
and where M has control measure mp, 0 < /? < 1, given in (8.4.3). It is easy to
verify (Exercise 8.7) that mp(Vt) < oo for all t > 0 and that the process X(t) =
M(Vt), t > 0, is ff-sssis with H = (3/a. Note also that m0(Vu n Vg nVt,)>0
for 0 < tj < t2 < *3 < co; so in view of (8.8.1), this process cannot have the
same finite-dimensional distributions as an (a, if)-Takenaka process.
Remarks
1. The argument presented in the preceding example shows that an (a,H)-
Takenaka process and the process defined in the example are different even
if these processes are defined for all ieM1.
2. One can construct other examples by setting Vt = Ct A Co where
Ct = Ct{ai,bi,a2,b2,... ,aN,bN)
= {{x, A)el'x E+: ai\ < x - t < 6jA, i=l,...,N},
where —oo < a\ < b\ < ai < bi < • ■ ■ < Ojv < few < oo (see Sato and
Takenaka(1991)).
3. While the spectral measure of any d-dimensional distribution of a general
Chentsov field is concentrated on 2(2d - 1) points (Theorem 8.5.1), that
of a Takenaka field defined on [0, co) is concentrated on d(d + 1) points
(Lemma 8.8.1). Observe that while 2(2d - 1) > d(d + 1), there is equality
when d = 2.
4. An (a, Jf)-Takenaka process as well as the process defined in
Example 8.8.2 are iJ-sssis Chentsov. They therefore have proportional two-
dimensional distributions (Theorem 8.6.2). Example 8.8.2 shows that their
three-dimensional distributions are different.
The following theorem states that the two-dimensional distributions of
Takenaka processes on [0, oo), in particular (a, i?)-Takenaka processes, determine all
the higher-dimensional distributions.

416
CHENTSOV RANDOM FIELDS
8.8
Theorem 8.8.3 Let 0 < a < 2 and let {X(t), t > 0} be a SaS Takenaka
process. Then for any d > 3 and distinct t\,ti,...,td, the distribution of
(X(t\),X(t2),..., X(td)) is determined by its two-dimensional marginal
distributions.
PROOF: Let m be the measure associated with X. Theorem 8.5.1 states that
the distribution of (X(ti),X(tj)) is characterized by the three numbers m(Vti D
Vtc.), m(Vtc. n Vtj) and m{Vti n Vtj). The proof of Lemma 8.8.1 shows that the
distribution of (X{ti),X(t2),--- ,X(td)), 0 < ti < t2 < ... < td, is determined
by m(RPtq), 1 < p < q < d, where the Rp,q are disjoint sets defined in (8.8.2).
It is therefore sufficient to prove that any rP)9 = m(Rptq) ,1 < p < q < d, can
be obtained from
aiJ = m(Vu nVtc.), bid = m(Vu n Vti), \<i<j<d.
Relation (8.8.2) implies
i j-i
^nyt' = U U ^ 1 < * < J < d.
p=l q=i
and thus
i j-i
<Hj = m{Vti n V*) = J]) J2 rp,„ l<i<j<d, (8.8.4)
p=I q=i
since the i?Pi9s are disjoint. For 1 < i < j < d, we obtain
* i-i
P=l P=l
i.e.,
rij = (ai,j+i - ai,j) ~ (a»-i,j+i - ai-i,j)i 1 < i < 3 < d.
When i = 1, Relation (8.8.4) yields
ri,j = ai,j+i - ai,j> 1 < 3 < d.
Similarly, when j = d,
i
Vttnvtd = \jRp,d, l <*<d,
p=i
and so bi>d = m(Vti nVtd) = J^p=l rPid, that is, rijd = bitd - &i_i,d, 1 < i < d,
andr))d = b\,d. ■

8.9
EXERCISES
417
The extension of the preceding result to non-Gaussian SaS Takenaka fields
on R1 and on R", n > 2, is more delicate.
Theorem 8.8.4 Let 0 < a < 2, n > 1 and let {X{t), t € W1} be a SaS
Takenaka field. Then, for any d > n + 2 and distinct t\,t2,...,td € Mn,
the distribution of(Xtt,Xt2,..., Xu) is determined by its (n + \)-dimensional
marginal distributions.
Proof: See Sato (19926) for n = 1,2 and Sato (1991) for general n. I
8.9 Exercises
Exercise 8.1 Prove that the integral (8.1.2) is well defined and that the
corresponding process has autocovariance (8.1.1). Express a in terms of a0.
Exercise 8.2 Consider d sets Bu..., Bd, let Zd be defined as in (8.2.9) and, for
6 e Ed) write B6k{k) = Bk if 6(k) = 1 and B6k{k) = the complement of Bk if
6{k) = 0. Show firstly that
B.U-UBd= U f]Bskw.
sezdk=\
Then prove
Bt = U *T
for j = 1,..., d.
Exercise 8.3 Evaluate mp(Vt) when n — 1 (see (8.4.6)).
Exercise 8.4 Let X£H and X™H be (a,if)-Takenaka fields on Rn and Rm,
respectively. Fix unit vectors en € R" and em € Rm and define the corresponding
projection processes {X£iH,en{u) = X£H(uen), u S R1} and {X™Hem(u) -
X™H(uem), u € R1}. Show that these processes are SaS, iJ-sssis Chentsov
and that there is a constant C such that -X"2,tf,en = CX™Hem in the sense of
equality of the two-dimensional distributions.
Exercise 8.5 Let {X{t) = M{Vt), t 6 R71} be a SaS Chentsov field with
associated measure to. Fix t = (ti,..., td) € (Rn)d and define Zd as in Exercise
8.2. For 6 e'Zd, let
„M . Vtk if*(*) = l, _ f Vt„ ifS(k) = I,
Vtck if 6{k) = 0, { Rd x K+ if 6{k) = 0,

418 CHENTSOV RANDOM FIELDS 8.9
V(t,6)=f]Vt[w, V(t,8)=f]Vtf\
k=l k=l
Td(S) = {6' = (<5'(1),..., S'(d)) e Sd: S'(k) > 6(k) for k = 1,..., d}.
Show that the associated measure m satisfies the following 2d — 2 equation
' £ m(V(t,6'))=rn(V(t,6))foT6e-Ld\{(l,...,l)}. (8.9.1)
This can be viewed as a system of 2d — 2 linear equations in 2d — 1 unknowns
m(V(t,(5)), 5 € 2d- The right-hand sides are the values of m(V(t, 6)), 8 e
Ed\{l,..., 1}, which are determined by the (d — 1)-dimensional marginal
distributions of (X{t,),..., X (td)).
Hint: See Sato (1991), p. 124.
Exercise 8.6 Write (8.9.1) in matrix form as Max = b where Md is a (2d - 2) x
(2d - 1) matrix of O's and l's. Let Md(k) be the (2d - 2) x (2d - 2) matrix
obtained from Md by deleting the kth column. Prove that Md(k) is invertible for
anyfc= l,2,...,2d- 1.
Hint: See Sato (1991).
Exercise 8.7 Verify the statements in Example 8.8.2.
Exercise 8,8 Consider the if-sssi process {X(t), t > 0} defined in Example
8.8.2. This process is not (a, i?)-Takenaka. Show, however, that the codifference
r{j) for the increment sequence Y(j) = X(j + 1) - X(j), j > 1, is still
asymptotically proportional to jaH~2 as j —> oo.

Chapter 9
Introduction to sample path
properties
We have defined a stochastic process {X(t), t € T} by its finite-dimensional
distributions. We now want to view it as a collection of random variables. Our
goal is to study its sample paths, i.e., the functions {X(t, u), t € T}, u e Q. We
shall do this in several chapters. In this chapter we introduce a number of basic
notions such as "versions," "separability" and "measurability," that are used in the
study of sample path behavior. We shall not assume at first that {X(t), t e T} is
stable because these notions apply to all stochastic processes. Some of the more
technical proofs in this chapter and the following ones can be omitted in the first
reading.
Two stochastic processes are versions of each other if they have the same
finite-dimensional distributions. We will say, for example, that {X(t), t 6 T} is
sample bounded if there is a version whose sample paths are bounded. Separable
versions, which are introduced in Section 9.2, allow the use of finite-dimensional
distributions to compute the probability of events involving uncountably many ts.
We use separability in Section 9.3 to obtain criteria for sample boundedness and
sample continuity in terms of finite-dimensional distributions. In Section 9.4 we
discuss measurability. Whereas separable versions of a stochastic process always
exist, there may be no measurable versions. We provide in this section necessary
and sufficient conditions for the existence of a measurable version. The conditions
involve the Condition S introduced in Section 3.11.
Section 9.5 is devoted to zero-one laws. Is it possible for a proportion 0 <
p < 1 of the sample paths to have a given property or must p be either 0 or 1?
Theorem 9.5.4 presents a very general zero-one law for stable processes that will
be used on several occasions in the sequel.

420
INTRODUCTION TO SAMPLE PATH PROPERTIES
9.1
9.1 Versions
In the preceding chapters, we characterized a stochastic process by its finite-
dimensional distributions. We now view a stochastic process X = {X(t), t € T}
as a collection of random variables defined on a probability space (Q, JF, P). Each
element u G Q. gives rise to a "sample path" or "realization," which is denoted
{X(t,cj),t€T}.
Definition 9.1.1 Two stochastic processes {X{t), t € T} and {Y{t), t e T}
are said to be versions of each other if they have the same finite-dimensional
distributions, i.e.,
{X(tn),n=l,...,N} = {Y{tn),n=l,...,N}
for any N and t\,..., t^ € T.
Hence a version of a stochastic process is a representative of the equivalence
class of all stochastic processes with a given set of finite-dimensional distributions.
Even if two versions of the same process are defined on the same probability space
they represent, in general, two different collections of measurable functions on
that space; their sample paths, in particular, may be quite different.
Example 9.1.2 Let e, e\, ej,. ■. be a sequence of i.i.d. exponential random
variables defined on a probability space (Q, jr, P). Define three stochastic processes
on (Q, T, P) indexed by t € [0, oo) as follows:
X(0) =0, X(t) = nife, + • • • + en < t < e, + --- + en+i, n = 0,1,2,...,
Y(0)=0, Y{t) = nif e, + --- + en < t < e, + • ••+ e„+i, n = 0,1,2,...,
f X(t) ift^e,
Z(t) = {
[ X(t) + 1 ift = e.
Observe that X is left-continuous, Y is right-continuous and Z is neither left- nor
right-continuous. But {X(t), t > 0}, [Y(t), t > 0} and {Z(t), t > 0} are
all versions of the same process, the standard time-homogeneous Poisson process
on [0, oo). Of course, only {X(t), t > 0} is a Poisson process as it is usually
defined, i.e., with right-continuous sample paths. It is useful to draw a typical
sample path of each of the three processes and to note the difference between the
pictures.
If {X(t), t e T} and {Y(t), t € T} are defined on the same probability
space and if for every t eT,
P(X(t) = Y(t))=l, (9.1.1)

9.2
SEPARABILITY
421
then obviously the two processes are versions of each other. Processes satisfying
(9.1.1) are typically called indistinguishable, but even indistinguishable processes
can have very different sample paths (the three processes in Example 9.1.2 are
indistinguishable but they have different sample paths).
9.2 Separability
Some versions of a stochastic process {X(t), t e T} have properties that facilitate
the study of sample path behavior. One such property is "separability."
Definition 9.2.1 A stochastic process {X(t), t £ T} on a probability space
(Q, J7, P) is called separable if there is a countable subset T* C T and an event
Q.q £ J- with P(Qo) = 0 such that for any closed set F C R we have
{u: X(t) £ F, Vt £ T*}\{u: X(t) 6 F, Vt £ T} C Do-
T* is called a separant for {X(t), t£T}.
This definition is useful if the parameter set T is uncountable. If the process is
separable, we may consider a countable subset T* of T and use finite-dimensional
distributions to compute probability of events, even those involving uncountably
many is. Note that separability does not imply "smoothness" of the paths. The
following example of a process consisting of i.i.d. random variables may be the
best illustration of this point. Such a process is, in a sense, the "least smooth"
one can think of, yet not only does it have a separable version but, in fact, every
version of this process is separable provided that the distribution of X{t) has full
support.
Example 9.2.2 Let X = {X(t), t £ T} be a stochastic process consisting
of i.i.d. random variables, i.e., for any t\,...,tn € T, the random variables
X(t\),..., X(tn) are i.i.d.. Assume that the distribution of X(t) is supported on
the whole of E, i.e., P(X(t) S (o, b)) > 0 for any -co < a < b < oo. If T is
countable, take T* = T. If T is uncountable, take T* to be any infinite countable
subset of T. Then any version of X is separable with separant T*. To verify this,
let
% = {u: {X(t,uj), t e T*} is dense inM}
oo ,
= nn u{<"WM-ri<-},
reQm=i ter-
where Q denotes the rationals. Clearly, P(£2o) = 0, since the support of X is
the whole of K. Choose, now, w e Q§, and let A = {X(t,u), t G T*} C F,

422 INTRODUCTION TO SAMPLE PATH PROPERTIES 9.2
where F is a closed subset in R. Since the set A is dense in R and its closure A
is also in F, it follows that F — R. Therefore, for any closed set F Q R, the set
{w: X(t) S F, Vi e T*}\{w: X(t) e F, Vt £ T} has probability zero because
it belongs to Qo if F ¥" ^ and it 's tne empty set if F = R. This proves that X is
separable with separant T*.
The definition of separability did not use any special structure of the parameter
set T. The set T, however, may have a special structure of which one can take
advantage. We will often suppose that (T, p) is a metric space with metric p; for
example, T can be the closed interval [0,1], the open interval (0,1), the positive
real line or a Euclidean space with the usual Euclidean distance. This permits us
to talk of (open) balls centered at y e T, i.e., By(r) = {x:p(x,y) < r}, and,
more generally, of open sets (a subset U of a metric space is open if for any y eU,
it contains a ball centered at y).
Sometimes, however, what is important are the open sets themselves and not
how they are generated. In these cases, it is not necessary to suppose that T is
metrizable, i.e., that T is a space whose open sets are generated by a distance p.
It is sufficient to suppose that (T, r) is a topological space. Recall that a topology
r on T is a collection of subsets of T, called "open," such that T and 0 are open,
the intersection of two open sets is open and the union (not necessarily countable)
of open sets is open. Any topology on T generates B(T), the Borel c-algebra on
T, which is the smallest cr-algebra generated by the open sets.
A base for the topology t is a subset of t such that every open set in r is the
union of open sets in the base. In a metric space, the (open) balls form a base for
the topology generated by the metric. Some bases are countable. For example,
in R, the set of open intervals (a, b) with a and b rational forms a countable base
for the topology generated by the Euclidean metric. In the following, we always
suppose that (T, t) is a topological space with a countable base.
Definition 9.2.3 Let (T, r) be a topological space with a countable base. A
stochastic process {X(t), t € T} on a probability space (Q.,T,P) is called
strongly separable in the topology r if there is a countable subset T* CT and an
event Qo € T with P(fio) = 0 such that for every I € r, every closed set F C R
{w: X(t) £ F, Vi G I n T*}\{w: X(t) € F, Vt e 1} C Q0-
T* is called a strong separant for {X{t), t € T}.
Remarks
1. Strong separability implies separability because T & r. Since £2o has
probability zero, separability allows us to replace events involving all is in

9.2
SEPARABILITY
423
T by events involving all is in the countable subset T*. Strong separability
allows us to go one step further and replace events involving all ts in any
open set I by events involving all is in the corresponding countable subset
IDT*.
2. The notion of separability is due to Doob (1953). Doob's "separability" is
called here "strong separability."
3. Separability and strong separability are defined here relative to the class
of closed sets because this is what we need, but these notions can also be
denned relative to other classes of sets, for example, the class of closed
intervals. Obviously, the larger the class, the more useful the (strong)
separability property (but fewer stochastic processes possess it).
The following proposition gives a useful criterion for strong separability.
Proposition 9.2.4 Let (T,t) be a topological space with a countable base. A
stochastic process {X{t), t € T} is strongly separable in the topology r if and
only if there is a countable dense subset T* C T and an event Qq £ T with
P(Oo) = 0 such that for every uj g Qq and every i 6 T, there is a sequence
{t„}£L, C T* such that tn -► t and X(tniw) — X{t,u>) as n — oo. T* is a
strong separantfor {X(t), t € T}.
PROOF: The sufficiency is obvious because if X(tn, uj) —> X(t, u) and X(tn, uj)
is in a closed set, then so is X(t, uj). To establish the necessity, assume that X is
strongly separable, and let T* be a strong separant for X and Qo the corresponding
exceptional set. Fix t 6 T and w £ CIq. Since r has a countable base, let Bn: n =
1,2,..., be the base sets containing t. Then /„ = fl™=i -^»> n = 1,2,..., are
open, ordered by inclusion and contain t. If Fn = {y: \y — X(t,uj)\ > ^}, then,
by strong separability, we cannot have X(s,uj) £ Fn for all s G InnT*. There
must therefore be a tn 6/„nf such that \X(tn,w) - X(t,u)\ < ~. Thus,
tn —* t andX(tn,uj) —> X(t,uj). I
In Exercises 9.1 and 9.3 we consider the existence of strongly separable
versions in two very important cases: (i) When {X(t), t € T} consists of i.i.d.
random variables and (ii) when {X(t), t 6 T} is continuous in probability.
As it turns out, every stochastic process has a separable version.
Theorem 9.2.5 Let {X(t), t S T} be a stochastic process defined on an arbitrary
probability space (Q, J7, P).
((') A separable version {Y(t), t € T} of{X(t), t e T} exists. Moreover,
{Y (t), t € T} can be defined on the same probability space {Q, T, P) in
such a way that P(X(t) = Y(t)) = 1 for every t € T.

424
INTRODUCTION TO SAMPLE PATH PROPERTIES
9.2
(;'/') Assume, in addition, that (T, r) is a topological space with countable base.
Then there is a version {Y(t), t e T} of{X(t), t € T), strongly separable
in the topology t, that takes values in the extended real line [—00, +00].
Moreover, {Y(t), t £ T} can be defined on the same probability space
(Q, T, P) in such a way that P(X(t) = Y(t)) = 1 for every t € T.
The proof of this theorem parallels the corresponding argument in Doob
(1953), but in a more general setting. Firstly, we present a few remarks.
Remarks
1. The theorem states that a strongly separable version of a given stochastic
process may take the infinite values ±00. We will see in the proof why such
infinite values appear.
2. The proof of part (ii) of Theorem 9.2.5 provides a construction of a strongly
separable version. Is there a different construction which provides a version
that takes only finite values? In general, the answer is in the negative (see
Exercise 9.4).
3. One should not be unduly concerned about the infinite values. Firstly, for
each t, P(\Y{t)\ < 00) = 1 anyway. Secondly, for "smooth" stochastic
processes (which are our main concern here) there is often a strongly
separable version that takes no infinite values. In fact, the "most irregular" of
stochastic process, the one consisting of i.i.d. random variables, admits a
strongly separable version that takes only finite values!
Lemma 9.2.6 Let {X(t), t € T} be a stochastic process on a probability space
(Q, T, P). Let Aobea countable collection of Borel subsets ofR and let A be the
collection of intersections of subfamilies of Aq. Then there is a countable subset
T* ofT such that, for every t S T, there is an event Qt e T with P(Qt) = 0
such that
{u)-.X(s) e A, Vs e T', X(t) ^}CQt
for every A £ A.
Proof: We start with the particular case where Aq consists of a single Borel set
A. Then, of course, A = {A} as well. For every countable subset C ofT, define
PC{A) = P(X(t) e A, Vi g C) (9.2.1)
and let a(A) = infc Pc{A). This infimum is achieved because there is a sequence
of countable subsets Cm of T, m = 1,2,..., such that Pcm (A) < a(A) + ^.

9.2
SEPARABILITY
425
Since PCm{A) [ a(A) as m T co and TJ = U„=l Cm, we have PTX(A) =
a(A), i.e., C = TA achieves the infimum. In the sequel, T*A continues to denote
a countable subset of T for which this infimum is achieved.
Now choose any t G T and let
at(A) = {w:X{s) £A,Vse TjJ, X(t) £ A}.
Since (9.2.1) with C = T*A U {i} yields
P(X(a) e A Vs € TX, X(t) eA)> a(A) = PrX(A),
we conclude
P(Qt(A)) = iV;(A) - P(X(S) 6A,VSe T£, X(t) G A) = 0.
This proves the statement of the lemma in the particular case of Aq consisting of
a single Borel set. To prove the general case where Aq is (at most) countable, fix
t G T and construct T*A and Qt(A) for each A e Aq. If T* = LUeA Ta and
Qt = DaeAo nt{A), then, clearly, P(Qt) = 0 for every t e T. Now, if B G A,
then B = f]kAk for some sequence Ak G A). & = 1)2,..., and
jw:X(s) G B, Vs G T*. X(t) g s}
= (j{^ *(*) € B, Vs G T*, X(t) $ Ak]
k
C |jf;:X(s) € 4fcl Vs G T*Ak, X(t) $ Ak)
k
c{Jnt{Ak)cnt,
k
as required. I
PROOF OF Theorem 9.2.5: Let Ao be the collection of all finite unions of closed
intervals with rational or infinite endpoints, and let A be defined as in Lemma
9.2.6. Observe that A is the collection of all closed sets in R.
(i) To establish the first part of the theorem, choose T* as in Lemma 9.2.6 and
define, for each t G T,
f X(t) if w g Qt)
Y(t) = {
{ X(to) if w G fit,
where fit is the null event described in Lemma 9.2.6 and to is any fixed point in
T*. Then {Y(t), t G T} is a separable process, T* is a separant and fio = 0.
Moreover, P(X{t) = Y(t)) = 1 for every t G T.

426 INTRODUCTION TO SAMPLE PATH PROPERTIES 9.2
(ii) To establish the second part of the theorem, let j3 denote a countable base of
(T, r). Apply Lemma 9.2.6 to conclude that for each B € /3, there is a countable
subset T*(B) of B with the following property: for every t e B there is an event
Of with P(Qf) = 0 such that
{X(t) e F, Vt € T*(B), X(t) £ F} C Q.f
for every closed subset F of E. Let T* = \JB€/3 T*(B), and for every t <E T, set
Ot = 0 if t e T* and Clt = UBe0 ^? if * 0 T". Since the base j3 is countable,
P(««) = 0.
Now define the process {Y(t), t 6 T} as follows: for each t € T,
( X(t) ifwgOt,
y(t) = __
lim .-.t X(s) if w 6 Qf
Obviously, Y(t) is a well-defined random variable (taking values in [—oo, +oo]),
and P(Y(t) ^ X(t)) < P(Q.t) = 0 for every t £ T. Hence Y is a version of the
process X defined on the same probability space as X.
To prove that {Y(t), t 6 T] is strongly separable, fix any / € r and any
closed subset F of [—oo, oo]. It is enough to show
{Y(t) eF,Vt€lDT*} = {Y(t) e F, Vi e I}. (9.2.2)
We will prove firstly that for t £ I, the set
At(J, F) = {Y(t) € F, Vt € J n T*, F(i) £ F}
is empty. We only need to consider the case t 0 T*. Choose any B € /? such
that t e B CI. Then At(J; F) C At(S; F). If w g £2*, then w £ £Jf, so that
w £ At(B; F). On the other hand, if w £ Q.t, then X(t) = Y"(t) 6 F for every
f S B n T* implies
Y(i) = fim —t X(s) = lim .-. Y(s) € F,
since F is closed. Therefore, again w & At(B; F). This means that At(B; F) = 0
and thus At(I; F) = 0. Since
{Y(t)eF, vteir\T*}\{Y(t)eF, vt e /} = (jAt(/;F) = 0,
46/
Relation (9.2.2) follows and we conclude that the process Y is strongly separable
with T* as a separant and the exceptional set Qq = 0. I

9.3
APPLICATIONS
427
9.3 Applications
Here, we consider sample boundedness, sample continuity and pointwise sample
continuity.
Definition 9.3.1 A stochastic process is said to possess a sample path property V
if there is a version of this process in which each sample path has the property V.
Example 9.3.2 A stochastic process {X(t), t'•€ T} is said to be sample bounded
if there is a version {Y{t), t e T} of {X(t), t € T} such that
{w: sup |Y(i) | =co} = 0.
A stochastic process {X (t), t € T} on a topological space (T, r) is called sample
continuous if there is a version {Y(t), t € T} of {X(t), t € T} such that
{o>: the map Y: T —» R is not continuous } = 0.
It is important to note that not every version of a sample continuous process
has all (or even "almost" all) continuous sample paths. Indeed, using the methods
of Example 9.1.2 it is easy to construct a version of Brownian motion {B (£), 0 <
t < 1} where all sample paths are discontinuous, i.e., {u: the map B: [0,1] —>
R is continuous } = 0! Note that according to Definition 9.3.1, Brownian motion
is then both sample continuous and sample discontinuous.
The study of sample continuity and other sample path properties requiring a
topology on T is often simpler when there is a metric p on T, that is, when (T, p)
is a metric space. The assumption "the topological space has a countable base" is
then equivalent to "(T, p) is a separable metric space." The meaning, here, of the
term "separable" is different from that in Definition 9.2.1.
Definition 9.3.3 A metric space (T, p) is separable if T has a countable dense
subset.
For example, the real line R with the usual Euclidean metric is separable
because the rational numbers are dense in R. If T is separable with a countable dense
subset {ti,t2i---}. tnen the balls Bti(r) = {a;: p(x,U) < r}, i = 1,2,... ,r
rational, form a countable base. Because most of the stochastic processes in
which we are interested are defined on separable metric spaces, we will frequently
assume that (T, p) is such a space.
We will also be interested in compact metric spaces (T, p), i.e., metric spaces
where every (infinite) sequence in T contains an (infinite) subsequence which
converges to a point in T. (The qualifier "infinite" is always subsumed.) The real

428
INTRODUCTION TO SAMPLE PATH PROPERTIES
9.3
line is obviously not compact (no subsequence of the integers converges in K), but
the extended real line [—00, +00] is compact and so is any closed interval [a, b] in
R.
The following proposition can simplify the search for a version in which all
sample paths are bounded and/or continuous.
Proposition 9.3.4 Let {X(t), t eT} be a stochastic process defined on a
probability space (£2, J-, P).
If T* is a separant for a separable version of {X{t), t £ T},then {X(t), t 6
T} is sample bounded if and only if
P(sup \X(t)\ <oo)= 1. (9.3.1)
teT-
If (T, p) is a compact separable metric space and T* is a strong separant for
a strongly separable version of{X(t), t € T}, then {X(t), t € T) is sample
continuous if and only if
P(X(t) is uniformly continuous on T*) = 1. (9.3.2)
Remarks
1. The statements (9.3.1) and (9.3.2) are well defined in the sense that they both
involve collections u£fl which are events (i.e., belong to J^. The
probabilities of these events, moreover, depend only on the finite-dimensional
distributions of {X(t), t € T}. (See Exercise 9.7.)
2. The version where all sample paths are bounded can be constructed as a
modification of the original separable version. It can be made separable
with the same T* as separant.
PROOF of Proposition 9.3.4: It follows from the first remark above that
sample boundedness (respectively, continuity) of {X(t), t S T} implies (9.3.1)
(respectively, (9.3.2)). On the other hand, suppose (9.3.1) holds, and let
(Y(i), t G T}, defined on a probability space {Q.', T'', P'), be a separable version
of {X(t), t € T} withT* as separant. LetA^ = {u/ € Q': supt6T. \Y(t)\ = 00},
and let Q.'0 be the null subset of the sample space Q.' which enters the definition of
separability of {Y(t), t € T}. Set £(, = A£, U ty. From the first remark above
and (9.3.1), conclude that P'(L'0) = 0. Notice that for any J £ 1^, one has
suptgT \Y(t)\ < 00. Now define for every t € T,
( Y(t) ifw'glj,,
Z(t) =
[ 0 if u/ € !(,.

9.3
APPLICATIONS
429
Every sample path of {Z(t), t £ T} is bounded and, obviously, {Z(t), t £ T} is
a version of {X(t), t £ T}. This proves that {X(t), t £ T} is sample bounded.
Similarly, (9.3.2) implies the sample continuity of {X(t), t £ T}. I
Continuity in probability is a pointwise property, whereas sample continuity is
a global one. Because a process which is not continuous in probability cannot be
sample continuous, we may want to assume from the outset that {X(t), t £ T}
is continuous in probability. In this case, we can choose T* to be any countable
dense subset of T (see Exercise 9.3). We then obtain the following modification
of the second part of Proposition 9.3.4. Here, the separable metric space (T, p)
can be assumed to be locally compact, i.e., every point of T is contained in some
open ball with compact closure. This allows (T, p), for example, to be a Euclidean
space.
Proposition 9.3.5 Let (T, p) be a locally compact separable metric space and let
T* be a countable dense subset ofT. Let {X(t), t € T} be a stochastic process
defined on a probability space (£2, T, P) which is continuous in probability. Then
{X(t), t £T} is sample continuous if and only if for every compact subset C of
T,
P(X(t) is uniformly continuous onT* f~l C) = 1. (9.3.3)
PROOF: Obviously, the sample continuity of {X(t), t £ T} implies (9.3.3). On
the other hand, suppose that (9.3.3) holds for every compact subset C ofT. We
know from Proposition 9.3.4 that for each compact C there is a sample continuous
version defined on C. Our aim is to construct a sample continuous version on T.
The Lindelof theorem (Kelley 1955) states that there is a countable subcover
of each open cover of a space whose topology has a countable base. Therefore
one can express T as a countable union of compact subsets such that each point t
of T belongs to the- interior of one of these subsets.
Now, let {Y(t), t £ T} be a strongly separable version of {X(t), t € T}
with T* as separant (Exercise 9.3), defined on (say) the same probability space
(n,.F, P). Conclude as in the proof of Proposition 9.3.4 that there are events
4n) € T with P{I^n)) = 0 for every n = 1,2,... such that for every w &
4n)- {yW> * e Bn} is uniformly continuous. Denote To = U~=i 4n)- T^"
F(Io) = 0. Define
f Y(t) ifwglo,
Z(t) =
[ 0 ifwelo,
t € T. It is obvious that {Z(t), t € T) is a version of {X(t), t £ T} and
that all realizations of {Z(t), t £ T) corresponding to w 6 Xo are continuous.
Fix uj # Zq . Since every point to £ T belongs to the interior of some Bn, the

430
INTRODUCTION TO SAMPLE PATH PROPERTIES
9.4
non-random function Z(t, u>), t € T, is continuous at to and hence is a continuous
function on T: Thus {Z(t), t € T} is a version of {X(t), t 6 T} with all sample
paths continuous, proving that {X(t), t e T} is sample continuous. I
A weaker property than sample path continuity is pointwise sample path
continuity.
Definition 9.3.6 A stochastic process {X(t), t € T} on a topological space
(T, r) is pointwise sample continuous if, for every to € 2~\ there is a version
{Y(i), i € T} of {X(t), t e T} such that every sample path of {Y(t), t € T} is
continuous at to- We say that {X(t), t € T} is sample continuous at each to 6 T.
A sample continuous stochastic process is obviously pointwise sample
continuous and if {X(t), t € T} is a non-random function on (T,r), then the two notions
are equivalent. When {X(t), t €. T} is random, the implication "pointwise
sample continuity => sample continuity" is false. Nevertheless, the implication holds
if we restrict ourselves to a certain class of stochastic processes. For example, it
holds for Gaussian processes on a compact metric space. Unfortunately, it does
not hold for stable processes on a compact metric space (Exercise 9.5). But, as we
will see later, pointwise sample continuity implies sample continuity for a certain
subclass of stable processes.
Here is a version of Proposition 9.3.4 tailored to pointwise sample continuity.
The proof is left as an exercise (Exercise 9.9).
Proposition 9.3.7 Let {X(t), t € T} be a stochastic process on a separable
metric space (T, p). Let to € T, and let T* be a strong separant for a strongly
separable version of {X(t), t S T}. Then {X(t), t € T} is sample continuous
at to if and only if
p( lim X(t) = X(t0) 1=1. (9.3.4)
\ t—to, t6T* /
9.4 Measurability
We have introduced separable versions of stochastic processes and proved in
Theorem 9.2.5 that every stochastic process possesses one. The situation is quite
different with measurable versions. We assume again that (T, p) is a separable
metric space.
Definition 9.4.1 A stochastic process {X(t), t £ T} on a probability space
(D, T, P) is called measurable if X(t, w):TxQ-*Kisa (jointly) measurable
function.

9.4
MEASURABILITY
431
It is often preferable to work with a measurable version of a given stochastic
process {X(t), t e T}. Unfortunately, not every stochastic process has a
measurable version. Necessary and sufficient conditions on the finite-dimensional
distributions of the process which ensure the existence of a measurable version
are given below. They are due to Hoffmann-J0rgensen (1973) and involve the
Condition S introduced in Section 3.11. Recall that {X(t), t 6 T} satisfies
Condition S if there is a countable subset To C T such that for every teT,
X{t) = plimfc_<00X(ifc) where tk e T0, k = 1,2,....
Theorem 9.4.2 Let {X(t), teT} be a stochastic process. Then {X(t), t e T}
has a measurable version if and only if the following conditions hold:
(1) {X(t), t e T} satisfies Condition S ;
(2) for every two Borel sets G and H and every s e T the map
t — P{X(s) e G, X{t) € H)
from T to [0,1] is measurable. Moreover, if(l) and (2) hold, then a measurable
version {Y(t), t € T} can be chosen to be strongly separable, defined on the
same space (Q.^.P) as {X(t), t € T} and satisfy P(X{t) = Y(t)) = I for
every teT.
Example 9.4.3 Consider the process {X(t), t £ [0,1]} comprising i.i.d. non-
degenerate random variables. As indicated in Section 3.11, this process does not
satisfy Condition S and hence by Theorem 9.4.2, does not have a measurable
version.
Remark. Condition S and condition (2) of Theorem 9.4.2 involve only two-
dimensional distributions. Hence only the two-dimensional distributions of
{X(t), teT} play a role in determining whether the process has a measurable
version. The one-dimensional distributions of {X(t), teT} are not sufficient
(Exercise 9.10).
The proof of Theorem 9.4.2 is based on the following proposition due to Cohn
(1972). To formulate it, we first need to introduce the Ky Fan distance. Let X
and Y be random variables defined on the same probability space. The Ky Fan
distance between X and Y is
a(X, Y) := inf {e > 0: P(\X - Y\ > e) < e}. (9.4.1)
This distance metrizes convergence in probability (i.e., Xn —* X in probability
as n —► oo if and only if a(Xn, X) —♦ 0 as n —» oo), and it converts the space
L°(Q) of random variables into a metric space (Exercise 9.11).

432 INTRODUCTION TO SAMPLE PATH PROPERTIES 9.4
Proposition 9.4.4 Let {X(t), t € T} be a stochastic process. Then {X(t), t €
T} has a measurable version if and only if the map t —* X(t) .from the metric space
(T, p) to the metric space (L°(Q), a), is (Borel) measurable and has separable
range, i.e., the range has a countable dense subset. In the latter case, a measurable
version {Y(t), t € T} can be chosen to be strongly separable, be defined on the
same probability space {Q.,T,P) as {X(t), t € T) and satisfy P(X(t) =
Y(t)) = 1 for each t€T.
PROOF: (i) Suppose firstly that the map t —> X(t) is measurable and has
separable range. Our goal is to construct a measurable version {Y(t), t € T}.
Fix n > 1. Because the range is separable, we can choose a collection
Ci(n), i — 1,2,... , of disjoint Borel sets in L°(Q) of a-diameter less than
2~n that covers the range {X(t), t € T}. The sets Bi(n) = {tG T:X(t) e
Ci(n)}, i = 1,2,... /constitute a Borel partition of T. For any two points
s, t e Bi{n), we have a(X(s), X{t)) < 2~n, implying
P(\X(s) - X(t)\ > 2-") < 2-" (9.4.2)
by the definition (9.4.1) of the Ky Fan distance a.
For each n and i choose tn^ € Bi(ri). Let Xn(t.uj) = X(tn,i,u) if
t e Bi(n), i = 1,2,..., n = 1,2,... . Each {X„{t), t € T}, n =
1,2,..., is a measurable process because for every Borel set A C E, we have
{(t,u):Xn(t,u) £A} = U,~i(Sj(n) x {w e Q:X(tnti,ij) € A}). It follows
from (9.4.2) that P(\Xn(t) - X(t)\ > 2"n) < 2~n for every t € T and n > 1.
Therefore, by the Borel-Cantelli lemma,
P( lim Xn{t) = X(t)) = 1 for every t e T. (9.4.3)
n—^oo
Now consider the measurable set E = {(£, w): Xn(t, uj) converges as n —» co}.
The stochastic process Xn, defined as Xn(t,w) = Xn(t,u) if (t,u>) 6 E and
Xn(t, w) = 0 otherwise, is measurable for every n = 1,2,..., and Xn{t,io)
converges as n —* oo for every t € T, w € Q-. Lety(t) = limn^ooXn(i), t € T.
Then {Y(i), t € T} is measurable as the limit of a sequence of measurable
functions and P{X{t) = Y(t)) = 1 for every t € T in view of (9.4.3). To show
that the version {Y(t), t € T} can be chosen to be strongly separable, one has to
modify slightly the last step above and use the measurable choice of limit points
lemma of Cohn (1972). We omit the details.
(ii) Suppose, now, that {X(t), t £ T} has a measurable version {Y(t), t € T}
defined on some probability space (£ii, -Fi.Pi). We want to show that the map
t —* X(t) is measurable and has a separable range. Without loss of generality,
we may assume that the probability space (fi^-T^Pi) is rich enough so that

9.4
MEASURABILITY
433
the following is true: for every random variable X0 G L°(£l) there is a random
variable Y0 G L°(Qi) such that {{X{t), t € T},X0} has the same finite-
dimensional distributions as {{Y(t), t G T},Yo}. Then for every e > 0, {t G
T: a{X{t),X0) < e} = {t € T: a(Y(t),Y0) < e}, and so it is enough to
prove that the map t —► Y(t), from (T, p) to (Z/°(£2i), a), is measurable and has
separable range.
Let M be the family of all measurable processes {Y(t), t G T} on
(i2i,^i,Pi) such that the map t -» Y(t) from (T,p) to (L0(Q,),a) is
measurable and has separable range. We want to show that M is closed under
pointwise convergence. More precisely, let Vfc, k = 1,2,... , be a sequence of
processes in M such that the sequence {Yk(t,uji)} converges as A; —► oo for each
t e T and iox G Qi. Let Y(t,wi) = lim^oo Yk(2,u;i). We must show that
7:, T x Q[ -> K is in M. Clearly, Y is measurable. Moreover, for each t G T,
as fc —> oo, Yfc(£) —> y(i) a.s. and thus in probability. The map t —* Y(t) from
(T,p) to (L°(<3i),a) is therefore the limit of a sequence of measurable maps
and is thus measurable itself. Moreover, denoting by Sj a countable subset of
(L°(Q.i), a) separating the range of t -» Yj(t), j = 1,2,..., we conclude that
Uj=i ■S'j separates the range of i —> Y(i). Therefore 7 e jM, and thus M is
closed under pointwise convergence.
Now let Ms be the family of "simple" maps Y from TxCl\ —> K of the form
Y(i,w,) = 2^1 ((*,«,) G \J{Aid x Bjj)),
2=1 j=l
where, respectively, Aij, Bj,j, j = 1,..., mi, i = 1,..., n, n = 1,2,..., are
measurable sets in £?(T) and .Fi. Clearly, Ms C M.
Since A4 contains A^s and is closed under pointwise convergence, we can use
a monotone class argument (see Exercise 9.13) to conclude that every measurable
process {Y(t), t G T} on (£l\,T\,P\) belongs to M. This completes the proof
of the proposition. I
Proof OF THEOREM 9.4.2: Suppose that {X(t), t G T} has a measurable version
{Y(t), t G T} defined on a probability space (Qi.^i.Pi). By Proposition 9.4.4,
the map t —» X(i) from T to (I/°(Q), a) has separable range, which implies that
{X(t), t eT} satisfies Condition S (as noted in the remarks following Definition
3.11.2 and established in Exercise 3.21). Moreover, for every two Borel sets G
and H and every set s G T,
P{X(s) G G, X(t) eH)= P(Y(s) G G, Y(t) G H)
= j l(y(S,Wl)6G)l(y(t,w1)6H)P,((iUl)

434 INTRODUCTION TO SAMPLE PATH PROPERTIES 9.5
is a measurable function of t by Fubini's theorem.
Suppose, on the other hand, that (1) and (2) of Theorem 9.4.2 hold. Let M
be the class of all bounded measurable functions /: RxI-tR such that for
any s € T, the map t -> Ef(X(s),X(t)) from T to R is measurable. Clearly,
M is closed under the pointwise convergence of uniformly bounded sequences
of functions. Let Ms be the family of "simple" functions /: R x K —♦ R of the
form
where AjjjBjj-, j = l,...,mj, i = l,...,n, n= 1,2,..., are Borel sets.
It follows from (2) that Ms Q M. We use Exercise 9.13 to conclude that M
contains every bounded measurable function /: R x R —» R. Taking, in particular,
f(x, y) = \x - y\ A 1, we see that the map t -> a(X(s), X(t)) from T to R is
measurable for any s e T where a(X, Y) = E(|X - y| A 1), X, y € L°(Q).
Note that 5 is a metric on L°(Q) equivalent to the Ky Fan metric a (see Exercise
9.12). Therefore, by (1), (i?(X),5) is separable, where X = {X(t), t € T} and
R(X) is the range of the map t -+ A"(t) in -L°(n).
Let R* be a countable dense subset of (H(X), 5) and let V be an open subset
of L°(£l). For every random variable Y & R* (IV, let
rK = sup {r > 0: VZ 6 L°(fl): 5(7, Z) < r =► Z € V}.
Since
H(X)nV= |J ' {Zeii(X):5(y,Z)<ry}I
refl-nv
we conclude that
{teT:X(t)eV}= (J {teT:a(X(sY),X(t))<rY}
YBR-nv
is a B(T)-measurable set. Here, sy is an element in T such that X(sy) = Y, Y 6
.R* n V. Therefore the map t —> X(t) from {T,p) to (L°(Q),5) is measurable
with separable range. Applying firstly Exercise 9.12 and then Proposition 9.4.4,
we conclude that the process {X(t), t e T} has a measurable version, which
can also be chosen to have the required properties. This completes the proof of
Theorem 9.4.2 I
9.5 Zero-one laws
In a previous section we discussed the meaning of some sample path properties
such as sample boundedness and sample continuity. The careful reader may have

9.5
ZERO-ONE LAWS
435
noticed that we have not discussed the possibility that such properties hold only
with some probability 0 < p < 1, for example, that only a proportion p of the
sample paths is sample bounded. This is because, as we will see below, for stable
processes, properties such as sample boundedness and sample continuity satisfy a
zero-one law; i.e., they hold either with probability 0 or with probability 1.
Definition 9.5.1 A stochastic process {X(t), t 6 T} is said to possess a sample
path property V with probability p > 0 if
p = sup {q > 0: there is a version {Y{t), t e T} of {X{t), t € T}
defined on a probability space (£2, T, P) and an event
Qo £ T with P(Qq) <l-q such that, for any u gO®,
the sample path Y(t, ui), t e T has the property V}.
It is obvious that Propositions 9.3.4 and 9.3.5 extend immediately to the
following:
Proposition 9.5.2 Let {X(t), t e T} be a stochastic process defined on a
probability space (Q, T, P) and let T* be a separant for a separable version of
{X(t), t e T}. Then {X(t), t € T} is sample bounded with probability p if and
only if
P(sup \X(t)\<oo)=p. (9.5.1)
Let (T, p) be a compact separable metric space and let T* be a strong separant
for a strongly separable version of {X(t), t € T}. Then {X(t), t € T) issample
continuous with probability p if and only if
P(X(t) is uniformly continuous on T*) ■= p. (9-5.2)
This proposition, whose proof is left to the reader, implies in particular that
the supremum over q in Definition 9.5.1 is achieved. We can now relate the
terminology introduced in the previous sections with the one we use here, in the
context of zero-one laws. Recall that by "{X{t), t € T} is sample bounded" we
mean "there is a version such that all paths {w: X(t,w), t € T} are bounded".
As a consequence of the previous proposition we obtain
Corollary 9.5.3 (i) A stochastic process {X(t), t e T} is sample bounded if and
only if it is sample bounded with probability 1.
(ii) Let (T, p) be a locally compact separable metric space. Then the process
{X(t), t € T} is sample continuous if and only if it is sample continuous with
probability 1.

436 INTRODUCTION TO SAMPLE PATH PROPERTIES 9.5
The next theorem presents a basic zero-one law for a-stable processes.
Because it is used on several occasions in the sequel, it is formulated in greater
generality than is required for sample boundedness and sample continuity. The
theorem involves a family M. of stochastic processes and a property V of the
sample paths. Because of the interplay between M and V, it is convenient to regard
explicitly each stochastic process as a function X: T x Q —♦ R of two variables
t eT and w € Q. X(t, •) is, as usual, Borel-measurable for each t € T. We shall
have to distinguish explicitly between the process X = {X(t,cj), t €T, u> e Q},
the function XU0:T x Q -+ R given by X(t,u) = X(t,uj0) for a fixed w0 € Q
and the realization X(-, ui0): T —> E where w0 € fi.
Theorem 9.5.4 Let (Q, JF, P) be a probability space. Let M. be a family of
stochastic processes
X = {X{t,2), t € T, 5 = (w,,W2,...) G Q.H}
defined on the product space (Q, T, P) = (D., T, P)N and depending on finitely
many coordinates1 of the product space, with the following properties:
(0 For every X and Y in M and any two real numbers a, b, the process
aX + bY is also in M.
(ii) If the stochastic process X is in M, then for every w0 € QN, X~ defined
above is also in M..
{Hi) If the stochastic process X = {X(t, u>), t € T, to e QN} is in M, then the
stochastic process X^ — {X(t, w(i)), t e T, to e QN} is also in M for
every i = 1,2,..., where for lj — (u)\, U2,...) € QN,
C(,) := (wi+i,wi+2,...).
Moreover, let V be a sample path property such that:
(rv) For every X in M the set {u>: X(-, w) has property V} is in T.
(v) IfX and Y are in MandX = Y, then
P(u>: X(■,&) has property V) = P(u>: Y(-,u>) has property V).
(yi) For every two functions f,g: T —> R with property V and any two real
numbers a, b, the function af + bg: T —♦ R has property V.
M may also include processes depending on zero coordinates, i.e., non-random processes.

9.5
ZERO-ONE LAWS
437
Then for any a-stable stochastic process X in M,
P(p: X(-, Hi) has property V) = 0 or 1.
Remark. The family M is denned independently of V. In some applications,
one may choose M to include all processes depending on any number of finitely
many coordinates, including zero coordinates. In that case, conditions (ii) and (iii)
are automatically satisfied. Moreover, because the probability space (Q, T, P) is
rich enough and because X depends only on a finite number of coordinates, it is
possible, using the algorithm in (iii), to construct as many independent copies of
X as we wish. These copies will all be in M.
PROOF OF Theorem 9.5.4: Assume, to the contrary, that
p := P(ui: X(;Z) has property V) € (0,1). (9.5.3)
Let X be an independent copy of X belonging to M. and depending on a different
set of coordinates. Then Y = X — X is a SaS process and is in M. Moreover,
q := P(5: Y(-,u) has property V) e (0,1) (9.5.4)
since q > p2 > 0 and 1 - q > 2p( 1 — p) > 0.
Our aim is to show that (9.5.4) cannot hold. Let Y be an independent copy
of Y belonging to M. and dependent on a still different set of coordinates. Then
using the fact that Y + Y — 2l/aY and the properties of M. and P, we obtain
q = P(w: Y{-,u>) has property V)
= P(uj: Y(-,uJ) + Y(-,u) has property V)
= P(p: Y{-,u>) + Y(-, u5) has property V, Y{-,U>) does not have it)
+ P[ui: Y(-,lj) has property V, Y(-,lj) has property V)
= P(uj: Y(-,u>) + Y(-,lj) has property V, Y(-,u) does not have it)
+ q2. (9.5.5)
It is enough to show
P(p: Y(-, w) + Y(-,u) has property V,Y(-,u) does not have property V) =0
(9.5.6)
since (9.5.5) and (9.5.6) imply q = q2, i.e., q = 0 or 1, contradicting (9.5.4).
Suppose that (9.5.6) does not hold. Then, by Fubini's theorem, there is a
function /: T —> R that does not have property V such that
P(2: Y(-,u) + f has property V) > 0. (9.5.7)

438
INTRODUCTION TO SAMPLE PATH PROPERTIES
9.5
Take any 0 £ (0,1), and let
Q.g = {w: Y(-, u>) + Of has property V).
The events D.g, 6 £ (0,1), are all disjoint. If this were not so, there would be
6\ j= 62 such that £2S| n Q«2 # 0- Then, taking lj £ Q.gt n Qg2, we obtain a
contradiction: on one hand (6i - 02)f = (Y(-,u>) + 0\f)- (Y(-,to) + 62f) has
property V by the properties of M and V, whereas on the other hand, the function
/ does not have property V. This shows that the events Qg, 6 £ (0,1), are all
disjoint.
Let Y be once again an independent copy of Y, in M, and dependent on a
different set of coordinates. Using the fact Y + 9f = BY + (1 - 9a)l/aY + Of
and the properties of M and V, we obtain
P(Qb) = P(2: 6Y(-,w) + (\- eaY'aY{-,u)+6f has property V)
> P(uj: Y(-,u) + / has property V, Y(-,u>) has property V)
= P(u: Y{-,U>) + / has property V) ■ P(w: Y(-,u) has property V),
which is positive by (9.5.4) and (9.5.7), implying that the probability P{Qg) is
positive and bounded away from 0. This is, of course, impossible, since there
are uncountably many of such Qes, all disjoint. Hence (9.5.6) must hold. This
completes the proof of the theorem. 1
Corollary 9.5.5 (i) An a-stable process {X(t), t £ T} cannot be sample
bounded with probability p, if 0 < p < 1.
O'O // (T, p) is a locally compact separable metric space, then an a-stable
process {X(t), t £ T} cannot be sample continuous with probability p if
0<p< 1.
PROOF: (i) Assume that X is sample bounded with probability p. Then
Proposition 9.5.2 implies that for some countable subset T* of T, p =
P(supteT. \X(t)\ < oo). Choose M to be the family of all stochastic processes
X indexed by T*, defined on the probability space (Q., T, P)N and depending only
on finitely many coordinates of the product space, including zero coordinates.
Observe that our process X belongs to M since it depends on one coordinate. Choose
V to be the property of boundedness of functions on T*. Conditions (i), (iii) and
(vi) of Theorem 9.5.4 are obviously satisfied. Condition (ii) holds because X~
depends on zero coordinates. Conditions (iv) and (vi) follow from the fact that
T* is countable. Applying Theorem 9.5.4, we conclude that p = 0 or 1.
(ii) To prove the second part of the corollary, suppose, to the contrary, that
{X(t), t £ T} is sample continuous with probability p £ (0,1). Let C be a

9.6
EXERCISES
439
compact subset of T. Then {X(i), t € C} is sample continuous with probability
q >p> 0. Our aim is to prove that q — 1.
To this end, we apply firstly Proposition 9.5.2 to conclude that there is a
countable subset C* of C such that
P(X(t) is uniformly continuous on C*) = q.
Then we take M. to be the family of all stochastic processes indexed by C*
defined on the probability space (Q, T, -P)N and depending only on finitely many
coordinates of the probability space. Choosing V to be the property of uniform
continuity for functions on C*, we can apply Theorem 9.5.4 and conclude that
q = 0 or 1. Since q > p > 0, we can only have q = 1.
To see that q = 1 leads to a contradiction, observe that if {X(t), t 6 C} is
sample continuous with probability 1 for any compact C C T, then it is continuous
in probability on T (continuity in probability is a pointwise property). Letting
T* be a countable dense subset of T and using Propositions 9.5.2 and 9.3.5 we
conclude that the process {X(t), t € T} is sample continuous, contradicting our
assumption. I
Remarks
1. There is no zero-one law that states that P(supn>l Xn < oo) = 0 or 1
for a family of jointly a-stable random variables X\, X2, ■ ■ ■ ■ (Take X\ to
be Sa{ 1,0,0) and set Xn = nXu n > 2. Then P(supn>1 Xn < 00) =
P(X\ < 0) = 1/2.) Therefore, there is no zero-one law for one-sided
sample boundedness of an a-stable process!
2. An argument similar to that of Corollary 9.5.5 can be applied to derive a
zero-one law for pointwise sample continuity; see Exercise 9.17.
9.6 Exercises
Exercise 9.1 As in Example 9.2.2, let {X(t), t G T} be a stochastic processes
consisting of i.i.d. random variables and assume that the support of the distribution
of X(t) is the whole of E. Let (T, r) be a topological space with a countable base.
Prove that every version of {X(t), t € T} is strongly separable in the topology
T.
Hint: If I 6 t is uncountable, choose as T* any countable dense subset of T such
that #(7 n T*) = co, where # denotes cardinality.
Exercise 9.2 Give an example to show that the statements in Example 9.2.2 and
in Exercise 9.1 are not generally true if the support of the distribution of X(t) is
a proper subset of R.

440
INTRODUCTION TO SAMPLE PATH PROPERTIES
9.6
Exercise 9.3 Let (T, r) be a topological space with a countable base and let
{X(t), t e T} be a stochastic process continuous in probability. Show directly
(i.e., without appealing to Theorem 9.2.5) how to construct a version of the process
{X(t), t € T} which is strongly separable in the topology r and whose separant
T* is a given countable dense subset of T.
Hint: For each ieT, there is a subsequence {tnic }£1, C T* such that X(tnk) —►
X(t) a.s. Define for each u> € Q, Y(t) = lim sup^^ X(tnk).
Exercise 9.4 Let Q. = [0,1], T = B, P = Lebesgue measure and define
( 0 w = f.
Show that every strongly separable version of the process {X(t), 0 < t < 1} has
to admit infinite values.
Hint: Let T* be the countable dense subset of [0,1] appearing in Proposition 9.2.4
and suppose without loss of generality that 1/2 e T*. Let {Y(t), 0 < t < 1} be
any strongly separable version of {X(t), 0 < t < 1} and let (Q', T'', P') be the
probability space on which it is defined. Set
A'={"'eQ'--W)-YTs)=t-S ^^^1-^^(0,1)^},
take J e Qqc n A' where P'(Q.'Q) = 0 and let t -+ | - p^j • Then use
Proposition 9.2.4.
Exercise 9.5 The purpose of this exercise and of the following one is to give a
feeling for the difference between continuity in probability, sample continuity and
pointwise sample continuity. At the same time, it provides a direct treatment of the
most important stable process-the SaS L6vy motion-without using the general
theory developed later in this chapter.
Let {X(t), t > 0} be a SaS Levy motion.
(a) Show that {X(t), t > 0} is continuous in probability.
(b) Let{rn}£L, C [0,t]. Show, using a proof similar to that of Kolmogorov's
inequality, that for any A > 0,
P{ max X(rn) > A) < 2P(X(t) > A).
n—\,...,N
Conclude that for each to > 0, {X(t), t > 0} is sample continuous at to-

9.6 EXERCISES 441
Hint: Let Q be the set of rational numbers, fix t0 e (0,1) and show that for
any A > 0 and 0 < e < min(io, 1 — *o)>
p( sup \X{t0) - X(r)\ >\)< 8P(X(e) > A).
(c) Use (b) to conclude that the sample paths of {X (t), t € [a, b]} are bounded
for any 0 < a < b < co.
(d) Forn = 1,2,..., i= 1,... ,2n, define
zf = sup \X(ri)-X(r2)\,
T-,,r2eQn(2-"(i-l), 2-"i)
(n)
and let Z* = maxi=i 2" Z\ '. Show that for any A > 0,
liminfP(Z: < A) < e-cx~""
n—*oo
for some C > 0. Conclude that P(limn_>00 Z^ — 0) = 0 and not 1, so that
the SaS Levy motion {X(t), t > 0} is not sample continuous.
(e) Show that {X(t),0 < t < 1} has a version in D([0,1]), the space of
functions on [0,1] that are right-continuous and have left limits.
Hint: See Breiman (1968), Theorem 14.27 and Corollary 14.29.
Exercise 9.6 Let Z\, Z2, ■ ■ ■ be i.i.d. SaS random variables, and let an [ 0 as
n->oo. Define {X{t), t e [0,1]} by X{±) = anZn, n = 1,2,..., X(0) =0,
and
X(t) = n(n+l)t(l(i)-l(^))+(n+l)x(^)-nl(l)
if l/(n+ 1) < t < 1/n.
Show that {X(t), t e [0,1]} is a SaS process and that it is continuous in
probability. Show that {X(t), t 6 [0,1]} is pointwise sample continuous if and
only if J^^Li a£ < 00. Thus, continuity in probability does not imply pointwise
sample continuity.
Exercise 9.7 Under the assumptions of Proposition 9.3.4, let
Abound = {w € Q: sup \X(t)\ < 00},
tev
Acom = {w £ fi: X(t) is uniformly continuous on T*}.
Show that Abound and Ac0nt belong to T and that P(Abound) and P(Acont) are
determined by the finite-dimensional distributions of {X(t), t e T}.

442 INTRODUCTION TO SAMPLE PATH PROPERTIES 9.6
Exercise 9.8 (i) Let {X(t), t e T} be a sample bounded stochastic process,
and let {Y(t), t € T} be a separable version of {X(t), t € T}, denned on a
probability space (Q, .F, P). Then there is an event Eo £ T with P(Xo) = 0 such
that for every w £ Xo, supteT \Y(t, u>)| < oo.
(ii) Let (T, p) be a compact separable metric space, and let {X(t), t e T}
be a sample continuous stochastic process. Let {Y(t), t e T} be a version of
{X(t), i6T} strongly separable in the topology r and defined on a probability
space (Q,.F,P). Then there is an event Xq e J" with P(£o) = 0 such that, for
every w £ So, Y: T —> R is continuous.
Exercise 9.9 Prove Proposition 9.3.7.
Exercise 9.10 Let /x be an arbitrary non-degenerate probability measure on the
real line. Let Xx = {X,(i), 0 < t < 1} and X2 = {X2{t), 0 < t < 1} be two
stochastic processes. Assume that P(X\(t) = ^(s)) = 1 for all s, t 6 [0,1],
whereas (^(ii),...,X2(tn)) are i.i.d. for any t\,..., tn € [0,1]. Assume also
that the one-dimensional distributions of the processes {X\(t), t € [0,1]} and
{X2(t), t € [0,1]} are identical and equal to /x. Show that {Xi(t), t € [0,1]}
has a measurable version, whereas {^(i), 0 < i < 1} does not.
Exercise 9.11 Show that the Ky Fan distance (9.4.1) is a metric on L°(Q) which
metrizes convergence in probability.
Exercise 9.12 Show that the functional
a(X,Y) = E(\X-Y\Al)
on L°(0) x L°(Q.) is a metric equivalent to the Ky Fan metric. ■
Exercise 9.13 Let (E, £) be a measurable space, and let £o be an algebra
generating £. Let Ms be the collection of all functions from E to M of the type
n
f(x) = Ylail(xeAi^
i=l
a, £R, A{ € So, i — 1,..., n, n — 1,2,.... Let M \ be a family of functions
from E to E that is closed under pointwise convergence and contains Ms-
(a) Let n > 1, au...,an £ R and A2,..., An be a partition of £ into £0-
sets. Let H be the family of all subsets H of E such that the function
/(x) = ail(x € #) + E"=2ai1(a; £ -M-ff) is in .Mi. Show that H is a
monotone class that contains £q. Conclude that H = £.

9.6
EXERCISES
443
(b) Show that every function of the form
n
f{x) = Y2ai*(x € A), x e E,
Oj € R, Ai £ £, i = 1,..., n, n > 1, belongs to M i.
(c) Conclude that Mi contains every measurable function /: E —> R.
(d) Show that if we assume only that M. \ is closed under pointwise convergence
of uniformly bounded sequences of functions (as opposed to assuming that
M.\ is closed under pointwise convergence), then M.\ contains all bounded
measurable functions /: E —» K.
Exercise 9.14 Let {X(t), t £ T} be a deterministic process on a separable metric
space (T, p), i.e., there is a deterministic function 77: T —> E such that for every
teT,
P(X(t) = »?(*)) = 1.
Show that {X(t), t € T} has a measurable version if and only if the function
77: T —+ R is measurable.
Exercise 9.15 (i) Show that every stochastic process {X(t), teT} continuous
in probability has a measurable version.
(ii) Show that every stochastic process {X(t), t £ T} with all sample paths
continuous is measurable.
Exercise 9.16 Prove Proposition 9.5.2.
Exercise 9.17 Let {X(t), t e T} be an a-stable process on a separable metric
space (T, p). Let to £ T. Prove that {X(t), t e T} cannot be sample continuous
at to with probability p, 0 < p < 1.
Exercise 9.18 In the course of the proof of Theorem 9.5.4 we have shown that
(9.5.4) and (9.5.7) cannot be simultaneously true for a jointly SaS sequence
{*n}£Li- Show that the following even stronger statement is true: (9.5.7) cannot
hold even by itself for a marginally SaS sequence {V,,}^.,.

Chapter 10
Boundedness, continuity and
oscillations
Stable processes display several types of sample path behavior. The a-stable
Levy motion, for example, has versions with right-continuous bounded paths.
Depending on the value of the self-similarity parameter, the linear fractional
stable motion has either versions with continuous paths or all its versions have
paths that are unbounded in any finite interval. We want to find convenient criteria
for such properties.
Let {X(t), t € T} be an a-stable process with integral representation
X(t) = JE f(t,x)M(dx), t s T. We will see that the sample path
properties of {X(t), t 6 T} are related to the smoothness of the non-random functions
{/(t.i), teT},xeE.
In the first part of this chapter we find conditions on / for sample boundedness
and sample continuity of a-stable processes. Applying the general criteria stated
in Proposition 9.3.4, we obtain in Sections 10.2 and 10.3 necessary conditions for
boundedness, continuity and pointwise continuity valid for any 0 < a < 2. In
Section 10.4 we show that these conditions are sufficient as well when 0 < a < 1
and also provide the precise asymptotic behavior of the distribution function
of the supremum (Theorem 10.4.2). The situation is more complicated when
1 < a < 2. The necessary conditions for boundedness are no longer sufficient
(Example 10.4.1), but the formula for the asymptotic behavior of the distribution
function of the supremum still holds for a > 1 provided that the process is a.s.
bounded (Theorem 10.5.1).
In the second part of the chapter we consider the oscillation process. This
process describes more delicate properties of sample paths than sample
boundedness or sample continuity. If the non-random functions {fit, x), t € T}, ie£,

446
BOUNDEDNESS, CONTINUITY AND OSCILLATIONS
10.1
in the integral representation of the stable process are uniformly continuous (the
precise assumption is stated in Condition 10.6.4), then the oscillation process is a
non-random upper-semi-continuous function with values in the extended interval
[0, oo]. That function takes values in the two-point set {0, oo} when 0 < a < 1.
We use the oscillation process to relate pointwise continuity to sample
continuity (Corollary 10.6.9). In Section 10.9, for example, we show that under
Condition 10.6.4, the sample paths of a stationary a-stable process or those of a
self-similar a-stable process are either a.s. continuous or a.s. unbounded.
The linear fractional stable motion however, does not satisfy Condition 10.6.4.
How general then is that condition? We prove in Section 10.11 that it is always
satisfied when the oscillation process is non-random and finite.
10.1 Introduction
Consider an a-stable process given in the form
X(t)= [ f(t,x)M(dx)+T](t),teT, (10.1.1)
Je
where M is an a-stable random measure on a measurable space (E, £) with control
measure m and skewness intensity /?, /(f, •), t € T, is a family of functions in
F (i.e., measurable functions satisfying (3.4.1) and (3.4.2)), and 77(f), t € T are
constants.
This representation is very general as we will see in Chapter 13. We shall
use it because it allows us to describe the conditions for sample path regularity
in a convenient way. We will see that the "regularity" properties of the family
of functions f{t,-), t e T, and of the family of constants 77(f), f € T, are
strongly related to the sample path regularity of the process {X(t), t € T} given
in (10.1.1). This may be surprising at first because a given a-stable process
{X(t), t eT} admits many different representations of the type (10.1.1).
We begin our discussion with an example illustrating the type of results we
hope to obtain.
Example 10.1.1 Consider the well-balanced symmetric linear fractional stable
motion
/oo
(|f - x\H-x'a - \x\"-1/a)M(dx), -00 < t < 00,
-00
defined in Example 3.6.5. Here, M is a SaS random measure on (R, B) with
Lebesgue control measure, 0 < a < 2, and 0 < H < 1, H ^ 1/a. Let us try
to guess the type of sample paths the process {^(f), f e R} posesses. We shall

10.2 NECESSARY CONDITIONS FOR SAMPLE BOUNDEDNESS 447
consider how the value of H affects the sample continuity of {X(i), f £l} for
a given a. Is {X(t), t € R} sample continuous for all possible choices of HI Is
it never sample continuous? Or, perhaps, is there a critical value of H, such that
the crossing of this critical value implies a radical change in the character of the
sample paths of our process (a kind of phase transition)?
Let us concentrate on the kernel
f(t,x) = \t- x\H-l/a - \x\"-1/Q, -co < t < oo, -oo < x < oo. (10.1.2)
Previously, we viewed this kernel as a family of functions in x, i.e., as /(£, •),
parameterized by —oo < t < oo. Let us now change perspective and regard it as
a family of functions in t, i.e., /(•, a;), parameterized by —oo < x < oo. Choose
any "typical" x, and thus regard f(t,x) in (10.1.2) as a function of one variable
t. It is a "nice" continuous function if H — 1/a > 0, but it becomes "less nice"
if H — 1J a < 0: it "explodes" at t = x. In the former case, the well-balanced
SaS linear fractional stable motion is an integral of a "nice" continuous function,
whereas in the latter case the process is an integral of a rather "badly behaved"
one. It turns out that this difference in the properties of the family of functions in
*i /(i x)> —°° < x < oo, is reflected in the properties of our linear fractional
stable motion {X(t), —oo < t < oo}. We will see in the sequel that the process
is sample continuous if H > 1/q and not sample continuous (in fact, much
"worse" than that) if H < 1 /a. Taking into account the range of our parameters,
0<#<l,0<a<2, we observe that the case H > 1/q is possible only if
a > 1. Therefore in the case 1 < a < 2, we will have a "phase transition" (an
abrupt change in the structure of sample paths) as H crosses the critical value of
1/a, whereas no such abrupt change occurs for 0 < a < 1, in which case our
process has "bad" sample paths for all possible values of H. (See Example 10.2.5,
Exercise 10.21, and Example 12.2.3.)
It should be noted that another "phase transition" occurs as a —> 2. Our
fractional process is perfectly well defined for q = 2 and 0 < H < 1, H ^ 1/2
(it is called fractional Brownian motion). But as can easily be shown, fractional
Brownian motion is sample continuous in all cases, whether H — 1/2 is positive
or negative! (Exercise 10.1).
10.2 Necessary conditions for sample boundedness
We start with the following result which, although subsumed by the more precise
estimates that are to follow, is useful in itself.

448 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.2
Proposition 10.2.1 Let {Xn, n > 1} be an a-stable process. If Z :=
supn>1 \Xn\ < oo a.s., then
fimA_00AaP(Z > A) < oo. (10.2.1)
Proof: Let {Xn, n > 1} be an independent copy of {Xn, n > 1} and let
Yn — Xn - Xn, n > 1. Then {Yn, n > 1} is a SaS process such that
W — sup,i;>1 \Yn\ < oo a.s.. Since, for arbitrary M > 0,
P(W>\) < P(sup|Xn|>A + M, sup|Xn|<M)
n>l n>l
= P(Z> X + M)P(Z<M),
we have iimA-cx>AQP(W > A) > )mix^00XaP(Z > X)P{Z < M). Letting
M —> oo we obtain
hrSA_00AQP(iy > A) > \^x^QO\aP{Z > A)
since Z is bounded. It is therefore sufficient to prove
lirnA_00AQP(W' > A) < oo. (10.2.2)
Let {Yn, n> 1} be an independent copy of {Yn, n > 1}. For any A > 0,
P(W>\) = P(sup|yn-Yn|>21/QA)
n>\
> 2P(\VX -W2> 2'/aA), (10.2.3)
where W\ and Wz are independent copies of W. As we will see, (10.2.3) is the
only property of W that is needed to prove (10.2.2).
Let A0 be any number such that P(W < 2'/QA0) > 3/4. For any A > A0, by
(10.2.3),
P(W>\) > 2P{WX > 21+1/QA) • P{W2 < 2i/aX)
> 3-P(W>2b\),
where6= 1+1/a. Thus, for every m > \,{\)mP(W > 2m6A0) < P(W > A0),
which implies
Imu_ooAb'P0y > A)<oo, (10.2.4)
where 6i = (log23-l)/(l + l/a). Since 6, < a, (10.2.4) is weaker than (10.2.2)
but it is an important first step. It implies £°1, P{W > C2j8/a) < oo for any
6 € (0,1) and any constant C > 0, or equivalently,
oo
Y[ P(W < C2je/a) > 0 (10.2.5)
j=k

10.2 NECESSARY CONDITIONS FOR SAMPLE BOUNDEDNIiSS 449
for some k (see Exercise 10.3).
Now let u0 > 0 be such that P(W < ug) > 0. Relation (10.2.3) implies
P{W > uo) > 2P(W > 21/au0 + ue0)P(W < ueQ). (10.2.6)
Defining um = 2l/aum-\ + u9m_u m> 1, we obtain after iterating (10.2.6),
P(W > um) < 2~m fW >Uo) 0 . (10.2.7)
Note that the limit I = lim™-^ 2~m/Qum exists, is non-negative and finite
(Exercise 10.2). Using
um ~ I2m'a as m -► oo (10.2.8)
and (10.2.5) we obtain
m—1 oo
Y[ P(W < u]) > C, JJ P(W < {le/2)2je/a) > 0
j=0 i=k>
for some k' > 1 and C\ G (0, oo). Therefore (10.2.7) gives
P(W > um) < C22-m, m> 1,
where C2 € (0,00). Using (10.2.8), we conclude that there is an integer m0 such
that
P{W > 2l2m'a) < C22~m, m > m0.
This of course implies (10.2.2) and concludes the proof. I
Corollary 10.2.2 If{Xn, n> 1} is ana-stable process such thatYimn^oo Xn =
0 a.s., then
lim iImA-.ooAQP(sup |Xn| > A) = 0.
*—P° n>i
Proof: Let Zi = supn>i |Xn| and Li = iirnA-,ooAQP(2i > A), i - 1,2,
Since Zi+i < Zi, Proposition 10.2.1 implies that {Lu i> 1} is a non-increasing
sequence of non-negative real numbers. We want to prove that L := limj_,oo£i =
0.
Suppose, to the contrary, that L > 0. Since lim^oo Z* = 0 a.s., for every
m — 1,2,..., there is an N{m) < 00 such that for every i > N(m),
P(Zi>2-m)<2-m. (10.2.9)
Assume, without loss of generality, that 1 < N(l) < N(2) < • • • , and let
y„ =mXn, if JV(m) <n<N(m+ 1), m = 0,1,2,...

450 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.2
withAT(O) = 1. Note that for every m = .1,2,..., maxJV(m_i)<n<jV(m) \Yn\ =
(m - l)maxjv(m_i)<n<jv(m)|-Xn|. < (m - l)ZAr(m_,) < Zjv(i) + -•• +
ZN(m-i) < J2T=i Z^U)- Hence-
oo
sup \Yn\ = sup max \Yn\ < V^ ZN(j) < oo a.s.
„>1 ' ' m>I JV(m-l)<n<JV(m) ^
by (10.2.9) and the Borel-Cantelli lemma. Therefore {Yn, n > 1} is an a.s.
bounded a-stable process. But for every m = 1,2,...,
nrHA_0OAaP(sup \Yn\ > A) > l5HA_0OAQP( sup \Yn\ > A)
n>\ n>N(m)
> limA-^ooA"P(mZN{m) > A) = maLN{m) > maL.
L > 0 yields lim^—ooAQP(supn>, |yn| > A) = oo, contradicting Proposition
10.2.1. Therefore L = 0. I
We now use Proposition 10.2.1 to obtain the following important necessary
condition for sample boundedness of a-stable processes.
Theorem 10.2.3 Let {X(t), t € T} be an a-stable process, 0 < a < 2, given
in the standard form (10.1.1). Let {Y(t), t € T} be a separable version of
{X(t), t € T} defined on a probability space (CI, T, P). Then
limA_ooAQP(supy(t)>A)>
sup { / h+(T*;x)a(l+P{x))m(dx) + f h-(T*;x)a(\-j3(x))m(dx)}
t-ct^Je Je '
(10.2.10)
where Ca is defined in (1.2.9), the supremum in the right-hand side of (10.2.10)
is taken over all countable subsets T* ofT and where
h+(T*;x)= sup{/(t,a;)}+, h.{T*;x) = sup{/(t,x)}_. (10.2.11)
teT" ter-
Moreover,
lMA-,ooAa-P(sup \Y(t)\ >\)>CQ sup / r(T*;x)am{dx), (10.2.12)
teT T'CtJe
where
f*(T*;x) = sup l/MI = max(/i+(r*;x), h_(T*;z)). (10.2.13)
t€T'

10.2 NECESSARY CONDITIONS FOR SAMPLE BOUNDEDNESS 451
In particular, if{X(t), t £ T} is sample bounded, then, necessarily,
sup / f*(T*;x)am(dx) < oo (10.2.14)
and
supteT |'/?(t)| < oo ifa^l,
supt6T \r](t) - ^ fEf(t,x)\n\f(t,x)\p(x)m(dx)\ < co ifa=l.
(10.2.15)
PROOF: Note firstly that by the separability of {Y(i), t £ T}, the left-hand sides
of (10.2.10) and (10.2.12) are well defined because supteT Y(t) andsupteT \Y(t)\
are well-defined random variables. Moreover,
supY(t) > sup Y(t) and sup|K(i)| > sup \Y(t)\.
teT ter- ter teT-
Hence (10.2.10) and (10.2.12) follow immediately from Theorem 4.4.5 and
Corollary 4.4.6, respectively. Of course, (10.2.14) now follows from Propositions 9.3.4
and 10.2.1 and from (10.2.12).
We now prove (10.2.15). The case a ^ 1 is easy. Let {X(t), t £ T} be an
independent copy of {X(t), t S T} and let
Z{t) = 2-"a(X{t) + X{t)) + (1 - 21-1/Q)r/(t)J t e T.
Then{Z(£), t e T} is aversion of {X(t), t £ T}, so it is sample bounded. Since
{X(t), t £ T} and {X(t), t e T} are also sample bounded, the above definition
of {Z(t), t £ T} shows that supteT |r?(i)| < co. Unfortunately, this "trick" does
not work for a = 1. We will consider this case separately.
Let a = 1 and suppose, conversely, that {X(t), t £ T} is sample bounded and
(10.2.15) does not hold. Then there is a sequence of points {tn, n = 1,2,...} C T
such that
lim
n—>oo
V(tn)-- / /(t„,x)ln|/(tn,a;)|/3(a;)TO(dx) =oo.
7T JE
Of course, we may assume, without loss of generality, that
lim an = co, (10.2.16)
n—*oo
where
an = V{tn) - - / /(in,x) In \f(tn,x)\/3{x)m{dx), n = 1,

452 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.2
LetX(tn) = X(tn)-an, n= 1,2,.... Then
P(sup|^(tn) + an| <oo) = 1
n>l
(10.2.17)
because {.X(i), t £ T} is sample bounded. Note thateachX(tn) has a distribution
of the type S\ (an, /3n, 0) with
Vn= \f{tn,x)\m(dx) < / sup \f(tn,x)\m(dx) := K <
JE JE n>l
oo
by (10.2.14). Therefore there is a finite constant -yK and a p > 0 such that
P(X(tn) > ik) > P for every n = 1,2,... (see Exercise 1.18). By (10.2.16),
there is for any A > 0 an n(A) such that a„(A) > A — yK. Then
P{suV\X(tn) + an\>\) > P(X(tnW)+anW>\)
n>l
> P(X(tn{X)) > lK)
> p-
Since A > 0 is arbitrary, we obtain P(supn>, \X(tn) + an\ = oo) > p,
contradicting (10.2.17) and thus proving (10.2.15) for a = 1. This completes the proof
of the theorem. I
Corollary 10.2.4 Let{X(t), t € M} be an a-stable process, 0 < a < 2. Suppose
there is a a countable subset T* ofR such that for every a < b we have
f
J a
f*(T*;x)m(dx) = oo.
(10.2.18)
there is an event ilo °f
(10.2.19)
Then for every version {Y(t),t 6 E} of {X{t),t €
probability 1 such that
sup \Y{t)\ - oo
a<t<b
for every w € £2o and every a < b.
We use the expression "the process {X(t), t 6 R} is unbounded on every
interval of positive length" to summarize the conclusion of the corollary
Example 10.2.5 Applying Corollary 10.2.4 to the symmetric well-balanced linear
fractional stable motion, we conclude that if H < 1 /a, the process is unbounded
on every interval of positive length. Indeed, for a given pair (a, b) take T* —
{r rational: a < r < b}. Since for every x G [a, b],
f*{T*;x) = sup \\t -x\H~''a - \x\H-^a\ = oo,
ter*
the necessary condition for sample boundedness (10.2.14) fails.

10.2 NECESSARY CONDITIONS FOR SAMPLE BOUNDEDNESS 453
Example 10.2.6 Applying the argument in Example 10.2.5 to the log-fractional
stable motion, we note that
f*{T*;x) = sup |ln|«-a;|-ln|a;|| = oo.
teT'
Therefore, the log-fractional stable motion is also unbounded on every interval of
positive length.
Example 10.2.7 Let 0 < a < 2, 0 < H < I/a and consider an (a,H)-
Takenaka field with n = 1 (see Definition 8.4.3), i.e., the //-sssi process {X(t) -
M(Vt), t 6 IR1}, where M is a SaS random measure with control measure
maH(dx,d\) = \a"-2dxd\, a: € K1, A > 0,
and where Vt is given in (8.4.2) and illustrated in Figure 8.2. As in the proof of
Lemma 8.4.2, for any T = {a < t < b} and any countable T* C T,
/oo /*oo
/ ( SUP Wt(x, X))amaH(dx,d\)
■oo JO t£T*
/oo /»oo
sup / lVt(x,X)maH(dx,dX)
■oo teT' Jo
1 f°°
= YZ^H J.^tfl \]xl°"~l ~ \x-t\aH~>\li\*M*-t\)dx = °°-
By Corollary 10.2.4, the Takenaka process {X(t), t S R1} is unbounded on every
interval of positive length.
It is obvious that there is always a countable fCT for which the supremum
in (10.2.10), (10.2.12) and (10.2.14) is achieved, and in many cases this T*
can be identified as follows. Suppose that there is a probability measure A on
{E,£) equivalent to the control measure m of the a-stable random measure
M in (10.1.1). (This is always the case if the control measure m is ^-finite; see
Exercise 10.4.) Regard (E, £, A) as a probability space on which random variables
f(t, -)s are defined. Then {/(£, •), t 6 T} is a stochastic process defined on that
probability space, and by Theorem 9.2.5 this process has a separable version
{9(t, 0. * S T} defined on the same probability space (E, £, A). We will call
{9(t, •), t G T} a separable kernel of the process {X(t), t € T) in (10.1.1).
Let T* be a separant for {g(t, •), t 6 T}. We claim that T* is the countable
subset for which we are looking. Indeed, for any other countable subset T,* C T,
A{a; S E:f(T*;x) < f*(T*;x)} = 0, so that
m(x e E: f*(T*;x) </*(!?;x)~) =0

454 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.2
as well. Hence,
/ r(T*;x)Qm(dx) > f r(T?;x)am(dx),
Je Je
as required. Similarly, the supremum in (10.2.10) and (10.2.12) is also achieved
onT*.
The preceding result also yields a necessary condition for sample boundedness
of complex-valued SaS processes given in the form
X{t)= I fc{t,x)Mc{dx), (10.2.20)
Je
where Mc is a complex-valued SaS random measure on (E, £) with circular
control measure k, and fc(t, ■), t e T, are complex-valued and in La(E, £, m),
where m is the control measure of M (i.e., m(A) = k(A x S2), A e £).
Corollary 10.2.8 Let {X(t), t € T} be a complex-valued SaS process given in
the form (10.2.20). If{X(t), t € T) is sample bounded, then
sup f f*(T*;x)am{dx)< 00, (10.2.21)
where
ft(T*;x) = sup |/c(t,x)| = sup l/j'Hi,*)2 + /c(2)(t,s)2|»/2, (10.2.22)
t&T" tgT*
and fc and fc are, respectively, the real and the imaginary parts of fc.
PROOF: Let V,(t) = Re X(t), Y2(t) = Im X(t). Then {Y^t), t e T} and
{^2(^)1 t € T} are real-valued SaS processes, which are both sample-bounded
if {X(t), t e T) is. By Proposition 6.2.4,
{Y(t), teT}=\ f g(t,(x,s))M(dx,ds), t€T}, (10.2.23)
where
g(t,(x,s)) = f^(t,x)Sl-f^(t,x)s2, (x,s)eExS2,
and M is a (real-valued) SaS random measure on {E x S2, £ x &) Wlth control
measure k. The sample boundedness of {Yi(£), t e T} implies, by Theorem
10.2.3,
sup // sup \f(ci\t,x)sl-fW(t,x)s2\ak(dx,ds)<oo, (10.2.24)

10.3 NECESSARY CONDITIONS FOR SAMPLE CONTINUITY 455
and, similarly, the sample boundedness of {Y2(t), t € T} implies
sup // sup \f(l)(t,x)s2 + rt2)(t,x)sl\ak(dx,ds)<oo. (10.2.25)
T
Hence,
sup / rc{T*;x)am{dx)
T'CtJe
= sup [ I sup\f^(t,x)2 + fP(t,x)2\a/2k(dx,ds)
T'CTJeJSi t6T*
< sup / / sup|/i,H*,s)s,-/i2>(t ,x)s2\ k(dx,ds)
+ sup / / sup \fW{t,x)s2 + f(2)(t,x)sl\ak(dx,ds)
T'CTJE Js2 <6T*
< 00.
This concludes the proof. I
10.3 Necessary conditions for sample continuity
Having obtained necessary conditions for sample boundedness, we now turn to
necessary conditions for sample continuity.
Theorem 10.3.1 Let (T, p) be a locally compact separable metric space, and let
{X(t), t € T} be an a-stable process, 0 < a < 2, given in the standard form
(10.1.1). Let T* be a countable dense subset ofT. If{X(t), t € T) is sample
continuous, then for every compact subset C ofT, we have
mix: f(t,x), t£T* D C is not uniformly continuousJ =0, (10.3.1)
/ sup \f(t,x)\am(dx) <oo (10.3.2)
Je t&T-nc
and
{r](t), t € T} is continuous if a ^ 1,
{rj(t) — 1 JE f(t, x) In |/(t, x)\P(x)m(dx), t e T} is continuous if<x — 1.
(10.3.3)

456 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.3
Proof: We start with our standard symmetrization argument. Let {X (t), t € T}
be an independent copy of {X(t), t 6 T} and define Y(t) = X(t)-X(t), t € T.
Then {Y(t), t 6 T} is a SaS process, is sample continuous since {X(t), t eT}
is sample continuous and has the integral representation
y(£)=2'/° / f(t,x)Mi(dx),teT,
where M\ is a SaS random measure on (E, £) with control measure m.
Note that sample continuity implies continuity in probability. Hence, if C is a
compact subset of T, then, by Proposition 9.3.5,
P(Y(t), t E T" n C is uniformly continuous) = 1. (10.3.4)
Observe that the control measure m is cr-finite on the support of each function
{f(t, x), x € E}, t e T* n C, and, therefore, it is cr-finite on the set E\ = {x e
^Eter-nc/^'^)2 > °}- Thus (10.3.1) is equivalent to
m(x € £7]: f(t,x), t € T* D C, is not uniformly continuous) = 0, (10.3.5)
and also
Y(t) = 2l/a f f{t, x)Mx (dx), teT*nC.
Je,
We shall show that (10.3.4) implies (10.3.5).
Let A be a probability measure on {E\, 8\) (where £\ is the restriction of £ to
£j), equivalent to the cr-finite measure m on (E\,£\). Let g(x) = ^(x) be the
corresponding Radon-Nikodym derivative, which can be chosen to take values
in (0, oo). Then on changing the variable of integration (Proposition 3.5.5), we
conclude that
{Y(t), teT*nC}~ [2l/a f f(t,x)g(xy'aM2(dx), teTfl c},
(10.3.6)
where M2 is a SaS random measure on (E\, £\) with control measure A. Now
we use the series representation of the integral in the right-hand side of (10.3.6)
(Theorem 3.10.1) to conclude that
oo
{Y(t), t€T*nC}± [da^iT-i/af(t,v4)s(Vi),/a, ternc},
(10.3.7)
where da is a constant, and the three independent series of random variables
are, as usual, a Rademacher sequence {ei,£2, • • •}» a sequence of arrival times
of a Poisson process with unit arrival rate {T\, f^,...} and a sequence of i.i.d.

10.3 NECESSARY CONDITIONS FOR SAMPLE CONTINUITY 457
i?!-valued random variables {Vi,V2,...} with common distribution A. Now
let {Z(t), t G T* n C} denote the stochastic process in the right-hand side of
(10.3.7). Then, by (10.3.4) and (10.3.7),
P(Z(t), t 6 T* n C, is uniformly continuous) = 1.
Let {Z(t), t G T* n C} be the stochastic process obtained by replacing e* by -et
fori = 2,3,..., on the right-hand side of (10.3.7). Since £1,-62, -«3>- ■ -is again
a Rademacher sequence, we have {Z(t), t G T* D C} = {Z(t), teT*C\C}
and thus
P(Z(t), teT*nC, is uniformly continuous) = 1
so
P(Z{t) + Z(t), feT'nC, is uniformly continuous) = 1.
But Z{t) + Z(t) =2daelr\/ag(V])i/af{t,Vi). Since g(Vx) e (0,00), we obtain
P{f(t, Vi), t G T* n C is uniformly continuous) = 1
or, equivalently,
X(x e Ei: f{t,x), t G T* n C, is not uniformly continuous) = 0.
This proves (10.3.5) because m and A are equivalent measures (on E\).
Relation (10.3.2) follows directly from the fact that sample continuity of
{X(t), t € C} implies sample boundedness of {X(t), t e C} and from Theorem
10.2.3.
It remains to prove (10.3.3). Once again, the argument for a ^ 1 is simple,
and is identical to the proof of the corresponding part of (10.2.15) in Theorem
10.2.3. This leaves us with the case a = 1. Denote
H(t) = V(t) - - [ f(t, x) In \f(t,x)\p(x)m(dx), t G T,
n Je
and suppose, to the contrary, that {X(t), t G T} is sample continuous but
{n(t), t G T} is discontinuous. Then there is a point t0 G T, an e > 0 and a
sequence of points {tn, n = 1,2,...} C T such that p(tn,t0) -» 0 as n -> 00 and
\n(tn) - fx(t0)\ > e for every n > 1. Clearly, sample continuity of {X(t), t G T}
implies
P( lim X(tn) = X(tQ)) = 1 (10.3.8)
xn—»oo
(Proposition 9.3.7). Let
Yn = (X(tn) - M(in)) - (X(to) " Kto)),

458 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.3
a„ = n(tn) - fJ.(to), n =1,2,.... Then, by (10.3.8),
1 = P( lim X(tn) = X(t0))
n—+00 '
= P( lim {Yn + an) = 0). (10.3.9)
The sequence {^(tn), n = 1,2,...} is bounded by Theorem 10.2.3. Therefore
we may (and will) assume that there is an L £ [e, oo) such that 0 < an —» L
as n —> oo. Note also that each Yn has the distribution Si(<7n,/?n,0) for some
an > 0 and — 1 < f3n < 1. If limn—oo an — 0, then, as n —► oo, Yn converges
in distribution to 0 and Yn + an to L, contradicting (10.3.9). Therefore all limit
points of the sequence {an, n > 1} must be non-zero. On the other hand,
let [Yn, n = 1,2,...} be an independent copy of {Yn, n = 1,2,...}. Then
Zn — Yn — Yn is a S|(2ct„,0,0) random variable, n > 1, and
P( lim Zn = 0) > P( lim (Yn + o„) = 0, lim (?„ + an) = 0) = 1
v n—»oo n—»oo n—>oo
by independence and (10.3.9). Hence, as n —* oo, Zn converges in distribution
to 0, implying limn^ooan = 0. This contradiction proves that {/i(i), t £ T}
must be continuous, and the proof of Theorem 10.3.1 is now complete. I
Theorem 10.3.1 can be extended to complex-valued SaS processes given in
the form (10.2.20). The proof of the following corollary is similar to the proof of
Corollary 10.2.8 and is left to the reader (Exercise 10.5).
Corollary 10.3.2 Let (T, p) be a locally compact separable metric space and let
{X(t), t £ T} be a complex-valued SaS process given in the form (10.2.20).
Let T* be a countable dense subset ofT. If{X(t), t € T} is sample continuous,
then for every compact subset CofTwe have
m(x: fc(t, x), t £T* C\C is not uniformly continuous) = 0 (10.3.10)
and
J sup \fc{t,x)\am(dx) <oo. (10.3.11)
JE t€T-nC
Let us now discuss the intuitive meaning of the results we have obtained,
namely Theorems 10.2.3 and 10.3.1. Assume that our a-stable process {X(t), t £
T} is given in the standard form (10.1.1) (take rj(i) = 0 for simplicity) and that
there is a probability measure A on {E, £) equivalent to the control measure m.
We can view {/(£, •), t £ T) as a stochastic process defined on the probability
space (E, £, A). Let (T, p) be a locally compact separable metric space. We know
from Theorem 9.2.5 that there is a strongly separable version {g(t, ■), t £ T} of

10.3 NECESSARY CONDITIONS FOR SAMPLE CONTINUITY 459
the stochastic process {/(£, •), t € T) which is defined on the same measurable
space (E,£) and such that \{x e E:g(t,x) jt= f(t,x)} = 0 for every t € T.
Since A is equivalent to m, we obtain
/ \f(t,x)-g(t,x)\am(dx) = 0
JE
and therefore
X{t) = / g{t,x)M(dx) a.s., f 6 T. (10.3.12)
JE
The representation (10.3.12) is called a strongly separable integral representation
of {X(i), t €T} and {#(£, ■), t S T} is called a strongly separable kernel.
Suppose, now, that {X(t), t e T} is sample continuous. Let T* be a
countable dense subset of T which is, at the same time, a strong separant for
{g(t, •), t € T}. Applying Theorem 10.3.1 and Proposition 9.3.4, we conclude
that for every compact subset C of T, {/(£, •), t e C} regarded as stochastic
process on the probability space (E1, £, X) is sample continuous. It is therefore
continuous in measure A and, thus, by Proposition 9.3.5, {/(*,-), t £ T} is sample
continuous.
We thus arrive at the following conclusion. Let {X(t), t eT}be an a-stable
process in a standard form (10.1.1). If {X(t), t € T} is sample continuous,
then {f(t, ■), t S T} is sample continuous as well. We remind the reader that
the sample path properties of the kernel in an integral representation are always
understood to hold with respect to a probability measure equivalent to the control
measure.
Of course, a similar argument can be used to show that an a-stable process
with a separable (not necessarily strongly separable) integral representation is
sample bounded only if the kernel {g(t, •), t € T} of this representation is sample
bounded. However, Theorem 10.2.3 yields actually a stronger result, namely not
only
m(x: sup \g(t, x)\ = oo) = 0
teT
but also
/ (sup \g(t,x)\)am(dx) < oo.
Je teT
Needless to say, none of the above results hold for Gaussian processes, i.e., when
a = 2 (see Example 10.1.1).
The following theorem gives necessary conditions for pointwise sample
continuity of a-stable processes.
Theorem 10.3.3 Let (T,p) be a separable metric space, {X(t), t € T} be an
a-stable process, 0 < a < 2, and let T* be a strong separant for a strongly

460 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.4
separable version of {X(t), t e T}. Suppose that {X(t), t E T} is sample
continuous at to € t. Then
■mix: either lim f{t,x) does not exist or lim f(t,x) ^ f(to,x)) =0
V t—,to t—to /
t€T" ter*
(10.3.13)
and in the cases a =fi 1 and a = 1, respectively,
!{r)(t), t € T} is continuous at to,
{r)(t) - I JE f(t, x) In \f(t, x)\l3(x)m{dx): t E T} is continuous at t0.
(10.3.14)
Suppose, in addition, that the point to has a relatively compact neighborhood,
i.e., a neighborhood whose closure is compact. Then for some open set A with
compact closure such that to 6 A C T,
I sup \f(t,x)\am{dx) <oo. (10.3.15)
JB teT'HA
The proof is very similar to the proof of Theorem 10.3.1 and is left to the
reader (Exercise 10.6).
The following example is a curious application of our results. It answers
the question posed in Section 6.6 by providing an example of a stationary SaS
process which is not conditionally stationary Gaussian.
Example 10.3.4 Let X(t) = L{t) - L(t - 1), 1 < t < 2, where {L(t), 0 <
t < 2} is the SaS motion. Clearly, X - {X{t), 1 < t < 2} is a stationary SaS
process. We claim that X cannot be conditionally stationary Gaussian.
Indeed, suppose to the contrary, that X is conditionally stationary Gaussian.
We know that {X(t), 0 < t < 2} is a.s. bounded (Exercise 9.5) and it follows
from Theorem 10.3.1 that {X(t), 0 < t < 2} is not sample continuous. Therefore
the conditional stationary Gaussian process must be sample bounded as well but
not sample continuous. But this violates Belyaev's theorem (1961) that every
stationary Gaussian process is either a.s. continuous or else a.s. unbounded on
every interval of positive length. Thus X cannot be conditionally stationary
Gaussian.
10.4 Necessary and sufficient conditions for sample
boundedness and continuity when 0 < a < 1
In the preceding section, we established necessary conditions for sample
boundedness and sample continuity of a-stable processes. The reader may wonder

10.4
NECESSARY AND SUFFICIENT CONDITIONS
461
whether these conditions are sufficient as well. We will show that the conditions
are sufficient if 0 < a < 1, but they are not if 1 < a < 2.
The following example demonstrates that the necessary condition (10.2.14)
for sample boundedness is not sufficient when 1 < a < 2.
Example 10.4.1 Let 1 < a < 2 and define a SaS process on the countable
parameter space
T={(N,A):N=1,2,---,AC{1,2,...,N}}
as follows: start with an arbitrary sequence {an, n > 1} of positive numbers
satisfying the following two conditions:
n=l
and
f((N,A),x) = <
JT<<oo, l<a<2, (10.4.1)
n=l
S^=i an|lnan| = co ifa=l,
(10.4.2)
Y^=\ an = °° ifl<Q<2.
(A possible choice is an = n~[ if a > 1 and an = (nlog2(n+ 1))_1 if a = 1.)
Set bn = an(n(n + 1))1/q, n = 1,2,..., and define the family of functions
{f((N,A),x), 0<x< 1}, (N,A) 6T,as
C bn ifxe (^-.^andne A
-bn ifx£(^,^],n<Nandn^A,
0 otherwise.
Consider the stochastic process
{X((N,A)) = J f{(N,A),x)M{dx), (N,A)€T},
where M is a SaS random measure on ((0,1), B) with Lebesgue control measure.
Clearly,
f\mxA),xWdx = j2bn(~^r[) = I2<<co
for every.(JV,A) € T, so {X({N,A)),(N,A) € T} is a well-defined SaS
process.

462 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.4
The necessary condition for sample boundedness (10.2.14) holds for this
process because, by taking T* = T, we obtain
f*(T;x)= sup \f((N,A),x)\ = bnifx€(^—,-}
and therefore
-1 oo , . oo
by (10.4.1). We will show nevertheless that
P( sup \X((N,A))\ = oo)=\, (10.4.3)
\N,A)£T
i.e., {X((N, A)), (N, A) e T} is not sample bounded.
To prove (10.4.3), define
ZB = ln(n+l)]-'/-Jlf((;riTfI])ln=l,2,....
Then Zi,Z2,...isa sequence of i.i.d. SaS random variables. Fix any w S Q, let
K> land 4.,= {n € {1,2,...,#}: Z„(w) > 0}. Clearly,
11
X{(KM){U) = J>|m((—,-])
n=l
K
= £an|Zn(o;)|.
n=l
Therefore, for any K > 1,
K
sup |X((JV,A))M|>X;an|ZnM|,
(Af.^)er n=l
and thus
sup |X((iV,A))|>Van|Zn|.
(W,^)6T ^
But Condition (10.4.2) implies Y^=\ an\Zn\ = oo a.s. (see Exercise 1.26),
establishing (10.4.3).
Exercise 10.7 describes a similar construction for the parameter space T —
[0,1], and also gives an example of a SaS process, 1 < a < 2, which satisfies

10.4
NECESSARY AND SUFFICIENT CONDITIONS
463
both (10.3.1) and (10.3.2) but is not sample continuous. We will encounter more
examples of this type later in the chapter.
The following is the main "positive" result of this section. It shows that in the
case 0 < a < 1, sample boundedness and sample continuity of a-stable processes
are completely determined by the corresponding properties of the kernel that
appears in their integral representation (see the discussion after Theorem 10.3.1).
Theorem 10.4.2 Let 0 < a < 1, and let {X(t), t €T} be an a-stable process
given in the standard form (10.1.1).
(i) Suppose that r){t) = Ofor every t € T in (10.1.1) and let T* C T be a
separantfor a separable version {Y(t), tsT} of{X{t), t £ T}. Suppose that
{Y(t), t e T} is defined on a probability space (Q, T, P). Then
lim AQP(suPr(t)> A)
A—oo tgy
= %•[/ h+{T*;x)a{l + P(x))m(dx) + f /i_(T*;i)a(l - f3{x))m{dx)
(10.4.4)
and
lim AaP(sup|y(t)| > A) = Ca ( r{T*;x)am(dx). (10.4.5)
A-.00 teT JB
In particular, {X(t), t 6 T}, with a general {r){t), t £T} in (10.1.1), is sample
bounded if and only if
L
/*(T*; x)am{dx) < oo (10.4.6)
E
and (10.2.15) holds.
(ii) Let (T, p) be a locally compact separable metric space. Then the process
{X(t), t 6 T} is sample continuous if and only if (10.3.3) holds and for every
compact subset C of T, for any countable dense subset C* ofC, we have
m(x e E: f{t,x), t € C*, is not uniformly continuous) — 0 (10.4.7)
and
[ sup \f(t,x)\am{dx) <oo. (10.4.8)
Je tec*
Proof: (i) In view of (10.2.10) and (10.2.12), the statements
rimA_00A0,P(supy(t) > A)

464 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.4
< %[ / h+(T*;x)a(l+p(x))m(dx) + [ h-(T*;x)a(\ -P{x))m(dx)
2 iJe je
(10.4.9)
and
rmA_ooAQP(sup|y(i)| > A) < Ca f f*{T*;x)am(dx) < oo (10.4.10)
teT je
will imply (10.4.4) and (10.4.5), respectively. We prove (10.4.9). (The proof of
(10.4.10) is similar; moreover, (10.4.5) can be deduced directly from (10.4.4); see
Exercise 10.8.)
Since there is nothing to prove if either one of the two integrals in the right-hand
side of (10.4.9) is infinite, assume
JBh+(T*;x)a(l +p(x))m(dx) < oo,
(10.4.11)
JBh-(T*;x)a(l - p(x))m(dx) < oo.
Let M\ and M2 be independent a-stable random measures on (E, S) with the
same control measure m (which is the control measure of M) and with the same
skewness intensity /3 — 1. Setting
W(t)= [ f(t,x){l+0(x))1/aMl(dx), t€T*,
Je
U(t) = / f(t,x)(l -P(x)y/aM2(dx), t € 7",
Je
we obtain
{Y(t), t€T*} = {2-x'a{W(t) - £/(*)), t e T*},
and, since {Y(t), t 6 T} is separable, we conclude
P(supy(t)>A) = P(supY(t)>\)
teT teT~
= P(sup(W{t)-U(t))>2l/aX)
teT-
< P( sup W{t) + sup (-U(t)) > 2'/QA).
teT- teT'
(10.4.12)
Set also
W. = / h+(T*;x)(l+(3(x))l/aMl(dx),
Je
U, = f h-(T*;x)(l-P(x)y/aM2(dx).
Je

10.4
NECESSARY AND SUFFICIENT CONDITIONS
465
By virtue of (10.4.11), W, and U* are well-defined a-stable random variables.
They are also independent, and it follows from Proposition 3.4.1 and Proposition
1.2.11 that for each t € T*,
W,-W(t)= f (/i+(T*;a;)-/(i)x))(l+/3(a;))1/aM1(dx)>0a.s.
Je
Therefore W* > supteT, W(t) a.s., and, similarly, U* > supteT. (-£/(£)) a.s.
We conclude from (10.4.12) that
ImiA_00AQP(y(i) > A) < lim \aP(Wt + U+ > 2'/QA)
A—*oo
CaU h+{T*;x)a{\ + {3{x))m{dx) + f h-{T*;x)a(l -/?(z))m(<£r)],
by Property 1.2.15 and the independence of W« and [/«. This proves (10.4.9).
The necessity of (10.4.6) and (10.2.15) follows from Theorem 10.2.3. On the
other hand, if (10.4.6) and (10.2.15) hold, then (10.4.5) implies
lim AQP(sup|y(i)|> A) <oo,
A—oo teT
and thus P(supt6T \Y(t)\ < oo) = 1 since 0 < a < 1. This proves that
{X(t), t e T} is sample bounded and completes the proof of the first part of the
theorem.
(ii) The necessity of (10.3.3), (10.4.7) and (10.4.8) for sample continuity of
{X(t), t G T} was established in Theorem 10.3.1. It remains to prove the
sufficiency of these conditions. We may and will assume -q(t) = 0 for every
teT.
We firstly show that the process {X(t), t € T} is continuous in probability
under (10.4.7) and (10.4.8). Let tQ £T and let B be a compact neighborhood of
t0. Let {tn, n = 1,2,...} C T be such that p(tn, t0) -* 0 as n -» oo. We may
assume that {tn, n = 1,2,...} C B. By (10.4.7), lirrin-.oo/Ctn.a:) = f(t0,x)
m-almost everywhere, and so by (10.4.8) and the dominated convergence theorem
lim / f(tn,x) - f(t0,x) m(dx) = 0.
n—°° Je
Proposition 3.5.1 implies that X{t„) —> X{t0) in probability as n -> oo and,
since t0 was chosen arbitrarily, we conclude that the process {X(t), t € T} is
continuous in probability.
In view of Propositions 9.3.4 and 9.3.5, it is enough to prove that {X(t), t £
C) is sample continuous for every compact subset C C T; we may therefore
assume without loss of generality that T itself is compact.

466 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.4
Let T* be a countable dense subset of T. According to Proposition 9.3.5, we
must prove
P({X(t), t e T*} is uniformly continuous) = 1. (10.4.13)
Set, as usual, f*(T*;x) = supter, |/(J,a:)| and define a new measure mi on
(E,£)by
mx{dx) = f{T*;x)am{dx).
Because of (10.4.8), mi is a finite measure. Let M\ be an a-stable random
measure on (E, £) with control measure mi and the same skewness intensity ft
as the random measure M. By Proposition 3.5.5,
{X(t),teT*}±{Y(t),t€T*},
where
Y(t) = / f(t,x)r(T*;x)-iMl(dx), t € T\
Je
(we adopt the convention 0/0 = 0.) Moreover, by Theorem 3.10.1,
{Y(t),teT*}±{CZ(t),teT*},
where C is a positive constant and
oo
Z(t) = ^7irr1/Q/(i,K-)r(T*; K)-1, t e T.
In the preceding expression, n, I"2,... is a sequence of arrival times of a Poisson
process with unit arrival rate; ( '), ( 2),... is a sequence of i.i.d. random vectors,
independent of the sequence Ti, T2,..., taking values infix {-1,1} and such
that Vi has distribution m = m]/m1(.E,)on£,andP(7i = ljV^) = (l+/?(Vi))/2.
We must therefore prove
P({Z(t), t G T*} is uniformly continuous) = 1. (10.4.14)
Set g(t,x) = f(t,x)f*(T*;x)-\ teT', x e E, and note that \g(t,x)\ < 1 for
every t and x. For t, s € T*,
oo
|Z(t)-Z(*)|<5]rr,/a|5(<,^)-s(*,Vi)|. (10.4.15)
t=I
Slightly abusing notation, denote by (D., T, P) the probability space in
which the stochastic process {Z(t), t £ T*} is defined (even though it is
{X(t), t e T} that was originally supposed to be defined on that space).

10.4
NECESSARY AND SUFFICIENT CONDITIONS
467
Introduce the subset Qi = {w G Q: limi_oori/i = 1 and {g{t, Vj), t G
T*} is uniformly continuous for each i = 1,2,...} and note that P(Qi) = 1 by
the strong law of large numbers and (10.4.7). In order to prove (10.4.14), it is
sufficient to show that {Z (t), t G T*} is uniformly continuous for every u> G Q \.
But the latter statement follows directly from (10.4.15) and the dominated
convergence theorem (the dominating factor is ci~lla for some c G (0,oo); note
that JX, i~xla < oo since 0 < a < 1). This proves (10.4.14). The proof of
Theorem 10.4.2 is now complete. I
Remark. Theorem 10.4.2 solves completely the problems of sample boundedness
and sample continuity for a-stable processes with 0 < a < 1 and gives the precise
asymptotic behavior of the distribution function of the supremum (of its separable
version). As we shall see, it is much more difficult to obtain analogous results
in the case 1 < a < 2, and there, different ideas are needed. This may seem
surprising at first, at least as far as sample boundedness and sample continuity are
concerned, because of the following argument.
Let {X(t), t € T} be, say, a SaS process with 1 < a < 2 and suppose that
we want to determine whether this process is sample bounded (continuous). Let A
be a Si/(2a) (1,1,0) random variable, independent of the process {X(t), t G T}.
Then
Y{t) = A1/aX(t), teT,
is a S^S process (see Proposition 3.8.1). Clearly, {X(t), t € T} is sample
bounded (continuous) if and only if {Y (t), t e T} is. Since {Y(t), t G T} is
Instable, we can apply Theorem 10.4.2 and determine precisely when it is sample
bounded (continuous). Hence, we know precisely what happens to {X(t), t G T}
as well!
Unfortunately, this procedure is not practical. The necessary and sufficient
conditions for sample boundedness (continuity) of {Y(t), t G T} are in terms
of its integral representation. We would be in good shape if we could obtain an
integral representation of {Y(i), t G T} which is explicitly related to "basic data"
involving the original process {X{t), teT} (say, to an integral representation
of {X(t), t G T}). But this seems to be very hard to obtain in general. We
are only aware of one general integral representation of the sub-stable process
{Y(i), t G T}, the one given in Proposition 3.8.2,
{Y(t), teT}=[c J X{t,u)M(duj), t G r}, (10.4.16)
where M is a S^S random measure on the probability space (£2,T, P) - the
probability space where the SaS process {X(t), t G T} is defined - and X(t, w)
is X evaluated at a fixedieTandweQ. Applying Theorem 10.4.2 to (10.4.16)

468
BOUNDEDNESS, CONTINUITY AND OSCILLATIONS
10.4
gives no information on sample path properties of {X(t), t e T} because all we
can conclude is that {X(t), t e T) is sample bounded (continuous) if and only if
{Y(t), t 6 T} is sample bounded (continuous) - a complete tautology!
Theorem 10.4.2 has a counterpart for complex-valued SaS processes.
Corollary 10.4.3 Let 0 < a < \,and let {X(t), t € T] be a complex-valued
SaS process given in the form (10.2.20).
(i) Let T* be a separantfor a separable version of {X(t), t € T}. Then the
process {X(t), t € T} is sample bounded if and only if
L
f*(T*;x)°m(dx) < oo, (10.4.17)
where f* (T*; x) is defined by (10.2.22).
(ii) Let (T, p) be a locally compact separable metric space. Then the process
{X(t), t € T} is sample continuous if and only if for every compact subset C of
T and for any countable dense subset C* ofC we have
m(x € E: fc(t, x), t € C* is not uniformly continuous) = 0 (10.4.18)
and
L
sup \fc(t,x)\am(dx) < oo. (10.4.19)
e tec~
PROOF: (i) The necessity of (10.4.17) has been proven in Corollary 10.3.2.
Suppose, now, that (10.4.17) holds. Let V, (t) = Re X(t), Y2(t) = Im X(t), t e
T. The processes {Yi(t), t £ T) and {Y2(t), t e T} are sample bounded by
(10.2.23) and Theorem 10.4.2. Therefore {X{t), t 6 T} is sample bounded. The
proof of (ii) is similar and is left to the reader (Exercise 10.9). I
Example 10.4.4 An immediate consequence of Corollary 10.4.3 is that, for 0 <
a < 1, every harmonizable SaS process (6.5.1) is sample continuous.
The situation in the case 1 < a < 2 is more complicated, as we will see in the
sequel.
Example 10.4.5 Consider the a-stable moving average
/oo
/(* - x) M(dx), t e E, (10.4.20)
-oo
where M is an ct-stable random measure on (R, B) with Lebesgue control measure
and skewness intensity fj, 0 < a < 1, and / is a measurable function satisfying
J — C
\f(x)\adx < oo. (10.4.21)

10.4
NECESSARY AND SUFFICIENT CONDITIONS
469
We will use Theorem 10.4.2 to show that the following is true:
The a-stable moving average defined in (10.4.20) with 0 < a < 1 is sample
bounded on an interval [a,b], -co < a < b < oo, if and only if for every
countable set T* C [0,1],
/
■OO
sup \f(t + x)\adx < oo. (10.4.22)
-oo ter*
It is sample continuous if and only if there is a continuous function g: R —* R
such that
Leb(a:: /(x) ^ g{x)) = 0 (10.4.23)
and
/oo
sup \g(t + x)\adx < oo. (10.4.24)
-oo0<t<I
Applying the first part of Theorem 10.4.2, we note that {X(t), a < t < b} is
sample bounded if and only if for every countable subset 5* of [a, b],
L
CO
sup \f{s-x)\a <oo. (10.4.25)
—oo s€S*
(See Theorem 10.2.3 and the subsequent discussion to reconcile (10.4.25) with
(10.4.6).) It is easy to show that (10.4.25) is equivalent to (10.4.22). Indeed,
suppose, for example, that b — a < 1 and let S* be a countable subset of [a, b].
Then S* - a is a countable subset of [0,1] and, if (10.4.22) holds,
/OO fOO
sup \f(s-x)\adx= / sup \f(t + x)\adx<
-oo sG5* J— oo t&S* — a
Suppose, on the other hand, that (10.4.25) holds. Then, as above,
oo.
/
oo
sup \f(t + x)\adx < oo
oo ter*
for every countable subset T* C [0, b - a}. Thus,
/
oo
sup \f(t + x)\adx < oo
oo t£T"+B
for every 9 > 0 and therefore (10.4.22) follows. The case b-a> 1 is similar.
For the sample continuity part of our claim, we apply the second part of
Theorem 10.4.2 and conclude that {X(t), t £ T} is sample continuous if and
only if for every closed interval [a, b], —oo < a < b < +oo, and for every
countable dense subset S* of [a, b]. Relation (10.4.25) holds and
Leb(cc eM.: f(t-x), t€ S* is not uniformly continuous) = 0. (10.4.26)

470 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.5
We have already proved that (10.4.25) is equivalent to (10.4.22). It follows from
(10.4.26) that there exists a continuous function </:!-»! such that (10.4.23)
holds. (The proof of this fact is an exercise in real analysis and is left to the reader
- see Exercise 10.10.) It follows immediately from (10.4.25) and (10.4.23) that g
satisfies (10.4.24).
On the other hand, suppose that a continuous g satisfying (10.4.23) and
(10.4.24) exists. Then (10.4.25) and (10.4.26) holds, and so {X(t), t G T}
is sample continuous.
The problem of pointwise sample continuity can be also fully solved in the
caseO < a < 1.
Theorem 10.4.6 Let 0 < a < 1 and suppose that {X{t), t € T} is an a-stable
process on a separable metric space (T,p). Let T* be a strong separant for
a strongly separable version of {X(t), t G T}. Let t0 G T have a relatively
compact neighborhood. Then {X(t), t G T} is sample continuous at t0 if and
only if (10.3J 3) holds, {rj(t), t £T} is continuous at t0 and for some open set A
with compact closure such that i0 G A CT, (10.3.15) holds.
The proof is left to the reader (Exercise 10.11).
10.5 Probability tails of suprema of bounded a-stable
processes, with index 0 < a < 2
Let {X(t), t G T} be an a-stable process given in the standard form (10.1.1), and
let {Y(t), t G T} be a separable version of {X(i), t G T}. For a G (0,1), the
asymptotic behavior of P(suptgT Y(t) > X) as A —> co is described in Theorem
10.4.2. For a G [1,2), however, we have only the asymptotic lower bound
given in Theorem 10.2.3. One may wonder whether either Relations (10.4.4) or
(10.4.5) continue to hold when a G [1,2). Example 10.4.1 shows that this is
not the case in general. Indeed, the a-stable process constructed in that example
is a.s. unbounded. It turns out that the difficulties are caused precisely by the
a.s. unboundedness of the paths. The following theorem shows that (10.4.4) and
(10.4.5) do hold when the paths are a.s. bounded. Its proof uses some results of
the theory of probability on Banach spaces.
Theorem 10.5.1 Let {X(t), t G T} be an a.s. bounded a-stable process, 0 <
a < 2, given in the form (10.1.1). Let T* be a separant for a separable version
{Y(t), t 6 T} of{X(t), t G T), defined on a probability space (Q, T, P). Then
(10.4.4) and (10.4.5) hold, i.e.,
lim AQP(suPy(t)>A)
A-»oo t&T

10.5
PROBABILITY TAILS OF SUPREMA
471
%[/ h+(T*;x)a{l + (3(x))m(dx) + f /i_(T*;x)Q(l - f3(x))m(dx)
*■ lJE JE
and
lim AQP(sup|y(i)| > A) = Ca [ f*{T*;x)am(dx),
where Ca is defined in (1.2.9), h+ and h-. in (10.2.11) and f* in (10.2.13).
PROOF: We start with the symmetric case, i.e., /? = 0 and 77 = 0. Let M be
a SaS random measure on (E, £) with control measure m given by fh(A) =
fAf*(T*;x)am(dx). Theorem 10.2.3 implies that m is a finite measure and
hence fh/fh(E) is a probability measure on E. Set g(t,x) = f(t,x)/f*(T*;x)
with the convention 0/0 = 0. Then
{Y(t), teT*} = if g(t,x)M(dx), teT*\
by Proposition 3.5.5 and {Y(t), t e T*} = {S(i), t e T*} by Corollary 3.10.4
where, in the notation of the corollary,
00
S(t) = (Cam{E))l'aY,«r;l/ag{t,Vi), t e T\
i=l
and VJ,, Vz, ■.. arei.i.d. E-valued random variables with distribution fh(-)/m(E).
Define the following stochastic processes:
00
sl(t) = (cam(£))1/a^£irl-1/ai(ri>i)9(i,yi),<er,
00
S2(t) = (Cam(E)y^Y,eiTiUai^ ^ l)9(t,Vi), t&T*.
Obviously, S{t) = Sx{t) + S2{t) for any t € T*. The processes {Si(t), teT*}
and {S2(t), t € T"*} are independent because the properties of Poisson random
measures ensure that {{eir~1/Ql(rj > l)g(t,Vi), t 6 T*}, i = 1,2,...} and
{{ejr~1/Ql(ri < l)g(t,Vi), t 6 T*}, i = 1,2,...} are independent Poisson
random measures on RT* (see, e.g., Resnick (1987), Section 3.3.2). Note also
that {S2(t), t € T*} is the sum of an a.s. finite number of bounded functions.
Therefore {SaW, i € T*} is an a.s. bounded stochastic process and, consequently,
{5i(i), t £ T*} is also a.s. bounded.
It follows from Proposition 2.7 of Rosinski (1990) that {S\ (t), t € T*} is an
a.s. bounded symmetric infinitely divisible process whose characteristic function
satisfies

472 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.5
■InEcxpi ^0(<)Si(*)
teT'
= m(E)-l.[ f fl -cos((Cam(E))]/au-,/a V 6(t)g{t,x))]m(dx)du
J\ Je1 V t^. 'J
(10.5.1)
for every {9(i), t G T*} with finitely many non-zero coordinates. We claim that
there is an e > 0 such that
£exp(e sup |5i(t)|) < oo. (10.5.2)
V tgT- '
To show this, identify T* with N = {1,2,...} for simplicity and take an arbitrary
sequence{a,, j > 1} of real numbers converging to zero. Let/?, = a,jS\(j), j =
1,2,... . Since Zj —> 0 a.s. as j -+ oo, the vector Z = {Z\,Z2, ■ ■ ■) can be
viewed as a symmetric infinitely divisible random vector taking values in the
separable Banach space cq of sequences of real numbers converging to zero with
the supremum norm || • ||oo, i.e., HZ^ = sup^, \Zj\. The dual (cq)' of Co is
the space /' of absolutely summable sequences (Reed and Simon (1972), Section
III.2). It follows from (10.5.1) and the lebesgue dominated convergence theorem
that for any 7 = (7!, 72,...) € /' = (co)',
oo
- In E exp 2(7, Z) = - In E exp j i ^ jjZj >
/ [l -cos(^(Cam(E)y/au-i/aYJljajg{j,x))]im(dx)du
J
Jen
(l-cos(7,y))F(dy), (10.5.3)
where F is the image of the measure fa{E)~x (Lebxm) on [1,00) x E under the
transformation <f>: [1,00) x E -+ cq given by
<Ku,x) = (cQm(E))l^u-l/a(aig(\,x),a2g(2,x),...)').
By virtue of (10.5.3), F is the L6vy measure of the symmetric infinitely divisible
random vector Z with values in cq. Since \g(j, x)\ < 1, we have for u > 1 and
x e E,
||^(u,s)||oo = (Cam(E)y/au-i/asnp\ajg(j,x)\ < (Cam(E)^a sup \aj\).
Therefore
F({y € cq: HylU > (CQm(£))'/Q sup K|})= 0,

10.5
PROBABILITY TAILS OF SUPREMA
473
i.e., the Levy measure F has bounded support. But de Acosta (1980), Corollary
3.3, shows that an infinitely divisible random random vector Z on a separable
Banach space with norm || • ||, whose Levy measure has bounded support, must
have all exponential moments finite, i.e., for any 6 > 0, i£exp(<5||Z||) < oo.
Since here Z = {a,jS\ (j), j > 1} is a vector in cq, we have, for every 8 > 0,
£expf5sup|aj||Si(j)|)< oo. (10.5.4)
Since (10.5.4) holds for every sequence {a.j, j = 1,2,...} converging to
zero, then (10.5.2) holds, because if (10.5.2) fails for every e > 0, we may
choose a sequence of positive integers Mm | oo as m -» oo such that
£;exp{imaxi<j<Mm |5i(j)|} > m. Then, setting a,- = \/m for Mm_, <
j < Mm, m = 1,2,..., Mo — 0, we obtain a sequence converging to zero for
which (10.5.4) fails with 5=1. This contradiction proves that (10.5.2) holds for
some e > 0.
We now turn to {S2 (t), t € T* }. The number of Tjs with values in the interval
(0,1] is a Poisson random variable N with mean 1, and given N, the I^s have
the same distribution as N i.i.d. i7(0,1) random variables arranged in increasing
order. Reasoning as in the proof of Theorem 1.4.2, we have the representation
N
{Sz(t), teT*} = (Cam(E))l^J2£iurX/a9(t, Vi), teT*, (10.5.5)
i=l
where N is Poisson with mean 1, independent of the sequences {ex, C2, - - •}
and {V\, Vz,...}, and where {U\, U2,...} is a sequence of i.i.d. 17(0,1) random
variables independent of the other random quantities appearing in the right-hand
side of (10.5.5).
For any A > 0, we have
P( sup S2(t) > A)
teT-
N
= P( sup (Cam(E)y^TeiU-l/ag(t,Vi)>\)
N
< p({Cam{E))x'a V sup nU7Uag(tM > >)
i=l
where Wi = (Cam(£))'/Q supteT. eiUr1/ag(t,Vi), i = 1,2,.... Set
/ h+{T*;x)am(dx)+ [ h-.(T*;x)am{dx)], (10.5.6)
H* = \Ca

474 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.5
and observe that
P(Wi>\) = \p([Camml/aU-X'ah^p^>\^
= X~aH*,
for A > (Cafh(E)y/a, because for such A,
■P\Ut<\ c°mW f.p.w*) - * Cam(E)E r{T..Vi)a
= \~aCa [ h±(T*;x)am(dx),
Je
since V* has distribution fh/fh(E) with m(dx) = f*(T*;x)am(dx). Use now
Exercise 10.12 to conclude
IIrn"A-*ooAQP( sup S2{t) > A) < H*. (10.5.7)
teT-
On the other hand, for any A > 0 and 0 < 6 < 1/2,
P( sup S2(t) > A)
teT'
= Ee~' ^p[ sup (^^(^))1/Q E *tff %(*. vo > a
oo , / n
> Efi_l-fP U{ sup(Cam(£7)),/'«6i£/r,/as(t, Vi) > A(l + 5),
—' Til \ . . weT"
„=o - V=i 'ter
\l/a,./-r-l/a, '• --*■ - ^
sup |(CQm(S))1^ejI/r'/«5(t, V})| < —— Vj = 1,... ,n, j ? i
t&T- n + l
n=0
xP( sup (Cafh(E))l/aeiU~1/ag(tM) > A(l +0)Y
MeT-
where <r(A) = P(supteT. |(CQm(£))1/Qeli71"1/Q5(t,y1)l < A). For A >
(CQm(£))'/<\
P(sup 52(t) > A) > ^e-'^-H*A-Q(l + 0)-[a(—-)1

10.5
PROBABILITY TAILS OF SUPREMA
475
Therefore,byFatou'slemma,liniA^c=oAQjP(suPt6T' s2{t) > A) > #*(l+0)-Q.
Since we may take 9 arbitrarily close to 0, we obtain, using (10.5.7),
lim AQP( sup S2{t) > A) = H*. (10.5.8)
A-*oo teT.
Our claim (10.4.4), in the symmetric case, follows now from (10.5.2), (10.5.6),
(10.5.8) and Lemma 4.4.2.
To prove (10.4.4) in general, let {Yi(t), t 6 T}, i = 1,2, be independent
copies of {Y(t), t € T}, and let Z(t) = Yx{t) - Y2(t), t € T. Then the process
{Z(t), t 6 T} is SaS and applying (10.4.4), we obtain
lim AaP(sup(y,(t) - Y2{t)) > A) = 2H*. (10.5.9)
From Theorem 10.2.3, we need only prove that
iimA_+00AQP(supy(t) > A)
< %[/ h+(T*; x)a(\ + (3(x))m(dx) + f h_{T*;x)a{l - /3{x))m{dx) .
2 LJE Je -I
(10.5.10)
Suppose, to the contrary, that there is a sequence Afc f oo as k —> oo such that for
some e > 0,
lim A£P(suPy(t)> Afc)
> Z.EL
- 9
+e.
[ h+{T*;x)a{l+(3{x))m{dx)+ f h_{T*;x)a{l -(3(x))m{dx)
■Je Je
Then for every M > 0,
limfc_oc(Afc - M)ap(suP(y,(t) - Y2(t)) >\k-M)
teT
> Um^ootAfc - M)ap(supy,(*) > Afcj suP|y2(t)| < m)
+ limfc_co(Afc -M)QP(sup(-y2<i)) > Afc, sup|Yi(t)| < M)
tex teT
> {2H* +c)P(sup |y(t)| < M).
teT
Since M can be taken arbitrarily large, we obtain a contradiction to (10.5.9). This
proves (10.5.10), and thus establishes (10.4.4). As we have seen before, this also
implies (10.4.5). The proof of the theorem is now complete. I
Remark. Theorem 10.5.1 follows from Theorem 4.4.5 when T = {1,... ,n}.

476 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.6
10.6 The oscillation process
We consider in this section a more delicate characteristic of the sample paths,
namely their oscillations.
Let {X(t), t € T} be a stochastic process on a locally compact separable
metric space (T, p) and let To be a strong separant for a strongly separable version
of{X(t), t£T}.
Definition 10.6.1 The stochastic process
Wx(t) = fiSSt,,*,-.*, t,,*26T0 \X{t2) - X{ti)\, t € T,
that takes values in [0, oo] is called the oscillation process of {X(t), t 6 T}.
We leave it to the reader to verify that different choices of T0 yield different
versions of the same oscillation process. In fact, much more is true (see the remark
following Property 10.6.10).
The following is an immediate consequence of the definition of the oscillation,
the properties of a strong separant (Proposition 9.2.4) and Propositions 9.3.4 and
9.3.7.
Proposition 10.6.2 For any fixed t0 € T, {X(t), t € T} is
(i) sample bounded at to if and only ifW(to) is finite a.s.,
(ii) sample continuous at to if and only if W(to) — 0 a.s.
It6 and Nisio (1968) proved that the oscillation process of any Gaussian
process continuous in probability is deterministic (non-random) in the following
sense. Let (Q, T, P) be the probability space on which such a process {X(i), t €
T} is defined. Then there is an event Qo C Q with P{Qo) = 0 and a (non-random)
function ax(t), t 6 T, such that Wx{t, u>) = ax{t) for every w € Q.q and every
t eT. This turns out to be false, in general, for stable processes.
Example 10.6.3 Let {L(t), 0 < t < 1} be the SaS motion. We know (Exercise
9.5) that this process is pointwise sample continuous and thus, by Proposition
10.6.2, Wx{t) = 0 a.s. for every t e [0,1]. Wx is not deterministic in the above
sense because if it were, we would have ax (t) = Oforallt e [0,1]. It would then
follow from Proposition 9.3.4 that the SaS motion is sample continuous which,
as we know, is not the case.
To characterize the a-stable processes for which the oscillation process is
deterministic, we will use their integral representation (10.1.1) rather than continuity
in probability.
Recall that, here, (T,p) is a locally compact separable metric space. We
consider the subclass of a-stable processes on T satisfying the following condition:

10.6
THE OSCILLATION PROCESS
477
Condition 10.6.4 {X(t), t e T} has a representation (10.1.1) with rj: T -> R
continuous and such that for every countable dense subset T* of T and every
compact subset C ofT,
m(x: the function /(■, x): T*nC->l is not uniformly continuous) = 0.
(10.6.1)
We will use the following technical result.
Lemma 10.6.5 An a-stable process {X(t), t € T} satisfies Condition 10.6.4
if and only if {X(t), t e T} has an integral representation (10.1.1) with the
following properties:
• 77: T —> R continuous,
• /(•, x): T —* R continuous for every x € E,
• there is a probability measure A on (E, £) equivalent to the control measure
m.
Proof: The sufficiency part is trivial. To prove the necessity, suppose that
{X(t), teT} satisfies (10.6.1). Let T* be a countable dense subset of T, and let
Eo = {x € E : for some t € T* and for some rational r > 0
such that the closed ball B(t, r) is compact,
the map /(-, x): T* fl B(t, r) -» R
is not uniformly continuous}.
Observe that m(Eo) — 0 and the limit lims_t, sgT- f{s,x) exists for every teT
and x e Eq. Therefore, the function g(-, x): T —* R, given by
is well defined and continuous for every x e E.
Fix any t e T and let E0{t) be defined as E0 but with T* U {t} replacing T*.
Then m(E0{t)) = 0. Clearly, for each x e E0{t)c,
f{t,x)= lim f{s,x)=g{t,x),
«—»t, sST"
and so for every teT,
m(xeE: f(t,x) ^ g{t,x)) =0.
Thus, a.s., for every teT,
X(t) = / p(t, x)M(dx) + V(t) = / s(t, i)M(da:) + »?(*),
7e Je+

478
BOUNDEDNESS, CONTINUITY AND OSCILLATIONS
10.6
where
E+=\J{xeE:\g(t,x)\^0}.
teT
Note that E+ G S since g(-, x): T —» R is continuous for every x G E, and thus
E+ — \JteT'{x e -^: IffC*'1)! ^ 0}- Moreover, m is tr-finite on E+ (Exercise
10.14). Using Exercise 10.4, conclude that there is a probability measure A on
(E+, £ D E+) equivalent torn. I
Wx(t) is the oscillation of X at a point t. We now define Wx(C), the
oscillation of X on a set C. This will allow us to overcome measurability
difficulties when dealing with {Wx (t), t 6 T}, which is generally an uncountable
set of random variables.
Definition 10.6.6 The oscillation of a process {X(t), t e T} on a set C C T is
where the limit superior is taken over all tj, tj G To such that d(t\, ti) —> 0 and
d(tuC)-*0.
It is easy to verify the following properties of the oscillation on a set. (If U
St
and V are random variables, we write U > V to indicate that U is stochastically
greater than V, i.e., P(U > A) > P(V > A) for all A.)
Proposition 10.6.7 (/) Wx({t}) = Wx{t) a.s.for every t G T.
O'O WaX(C) = |a|Wx(C) a.s./or every rea/ a,CCT.
(Hi) Wx+y(C) < WX{C) + WY(C) a.s.for every C C T, where {X(t), t G
T} and {Y(t), t £T} are stochastic processes defined on the same
probability space.
(iv) Suppose additionally in (Hi) that the set Cis compact and that {Y(t), t G T}
is sample continuous. Then Wx+y{C) = Wx{C) a.s.
(v) Suppose additionally in (Hi) that the set C is compact, and that {Y(t), t G
T) is independent of{X(t), teT} and continuous in probability. Then
Wx+y(C) > WX(C).
PROOF: We prove only (v) the rest is left to the reader. See Exercise 10.15.
We can assume that {X(t), t G T} is defined on the probability space
(QuT\,P\), and {Y(t), t G T} is defined on a different probability space
(Q2,^*2,-P2), where CT is the cr-algebra of cylinder sets. We view the three
stochastic processes {X(t), t G T}, {Y(t), t G T} and {X(t) + Y(t), teT} as

10.6
THE OSCILLATION PROCESS
479
defined on the product probability space (£2, x Q2, T\ x F2, P\ x P2). Let T0 be
a common strong separant for a strongly separable version of {X(t), t G i } and
a strongly separable version of {F(t), t G T}. Let WX(C) and W^n-vCC) be
defined by the same T0 and regard the random vector (Wx (C), Wx+y(Q) as an
element of the product space. The set A = .{(un,<jj2) G £2, x £22: Wx+y(C) >
Wx(C)} is then in fixf2.
In order to establish the result, we prove (Pi x Pi){A) = 1 by using Fubini's
theorem as follows: for a fixed wj € Qj, there are two sequences depending on
wi, {i£°, n = 1,2,...} C T0, z = 1,2, and a point i0 S C such that as n -> oo,
*n —► to, i = 1,2, and
lim |X(4'))-X(42))| = Wx(C).
n—»oo '
Since {Y(t), t G T} is continuous in probability at t0, there arc subsequences
{&l, k = 1,2,...},» = 1,2, and an event A>, G J"i with P2(A„) = 1 such
thatlimfc^ooy^.wz) = y(*o,W2), * = 1-2, for every w2 G /W Clearly,
(w,,w2) G A for any w2 e A^,,. Thus /n2 lj4(o;i,w2)P2(^2) = 1 for all
w, G Hi, which implies (P, x P2)(vl) = 1. ■
We shall now use the notion of oscillation on sets to derive sufficient conditions
for the oscillation process {Wx{t), t G T} to be non-random.
Theorem 10.6.8 Let (T, p) be a locally compact separable metric space and
let {X{t), t eT} be an a-stable process satisfying Condition 10.6.4. Then the
oscillation process { Wx (t),teT}is non-random, i.e., there is an event QqQQ-
with P(flo) = 0anda deterministic function ax: T - K+ U {oo} such that
Wx (t) = ax (t) for every u> g Qq and every t £T.
PROOF: Our argument depends only on the distributional properties of the
oscillations WX(C) on compact sets C, which are the same for all versions of
{X(t), t G T}. We may therefore assume, without loss of generality that the
representation (10.1.1) is of the form prescribed by Lemma 10.6.5 and that 77 (i) = 0.
Let A be a probability measure on (E, S) equivalent to the control measure m and
let
We suppose that a ^ 1 as the case a = 1 is similar to the case 1 < a <
2 considered below. Using Proposition 3.5.5 and Theorem 3.10.1, we obtain
{X(t), t€T} = {cY(t), t G T} where c> 0 and
CO /-
Y(t) = X;[7irrI/ar(V;)/(«, V0 - &|a) JEr{x)f(t,x)l3{x)X{dx)

480 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.6
As usual, n, Tz, ■ ■ ■ is the sequence of arrival times of a Poisson process with
unit rate, (^j), (^),... is a sequence of i.i.d. random vectors such that V* has the
distribution A on E,
i+m)
i>(7i = l|vri) = l-P(7,- = -l|v,i)
(/3(-) is the skewness intensity of the random measure M in (10.1.1)) and the
two random sequences are independent. The constants b\a' are given in Theorem
3.9.1.
Let CCTbea compact set. We firstly show that the oscillation Wx(C) is
non-random.
The case 0 < a < 1 is particularly straightforward because b\a' = 0 for every
i and Proposition 10.6.7 shows that for every N > 1, Wy{C) = WyN{C) a.s.,
where
oo
YN(t) = £ 7irr1/ar(Vi)/(i) v;), t e T.
i-N+\
Let ei,e2,.-. be i.i.d. standard exponential random variables such that I\ =
d + • • • + ej, i = 1) 2, • • • . Let -k be an arbitrary permutation of the indices
{l,2,---,iV}and
N
Y*{t) = ]T>-(0(e*a) + • • • + ewW)-'/ar(Vw(i))/(t,yTW) + yN(t), t € T.
t=i
Applying Proposition 10.6.7, we obtain Wy(C) = WYn(C) a.s. Hence,
WY* (C) = Wy (C) a.s. for any N > 1 and any permutation 7r of {1,2,..., N}.
The Hewitt-Savage zero-one law applies and yields
WX(C) = \c\WY(C) = constant =: ax{C) a.s. (10.6.2)
We now turn to the easel < a < 2. Letg(t) = fEr(x)f{t,x)P(x)\(dx), t €
T. If for each teC,
g(t) = lim ff(tn), (10.6.3)
then the same argument as in the case 0 < a < 1 applies, and (10.6.2) still holds.
Suppose, now, that (10.6.3) is violated at a point t0 € C. Consider a closed
ball B{t0,r), r > 0, and the integral
Jr(*o)= / sup |/(t,x)|am(cte).
J£ t€TonJ3(t0,r)
If Jr(t0) < co for some r > 0, then (10.6.3) holds at t0 by the dominated
convergence theorem. Thus JT(to) = ooforallr > 0. Applying Theorem 10.2.3,

10.6
THE OSCILLATION PROCESS
481
we conclude that {Y(t), t€T0n B{t0, r)} is not a.s. bounded for any r > 0 and
hence, by the zero-one law (Theorem 9.5.4),
P(w: sup \Y(t)\ = oo for all r > o) = 1.
This implies WY(t0) = oo a.s., so (10.6.2) still holds (with an infinite constant).
We shall use (10.6.2) to conclude the proof of the theorem.
Let A be the collection of finite intersections of closed balls of the type B{t, q),
where t € T0, q > 0 is rational and such that B(t, q) is compact. Since A is a.
countable collection of compact sets,
P(WX{C) = ax{C) for every C € A) = 1. (10.6.4)
Let Q% denote the event in (10.6.4). Fixing w € Qq and * 6 T* choose r > °
such that B(t,r) is compact, and choose a sequence {£„, n= 1,2,...} C To
such that p{tn,t) < r2~(n+i\ n = 1,2,... . Choose also rational numbers
qn, 11=1,2,..., such that p(tn,t) < qn < 2p(t„,t). Then
n
j=l
is a decreasing sequence offsets, and {i} = fT=, C- Hence (Exercise 10'16)'
Wx(t) = lim Wx(Cn), (10-6-5)
n—oo
which, of course, does not involve w e Ǥ. This completes the proof of the
theorem. I
The following is an immediate and important corollary of Theorem 10.6.8.
Corollary 10.6.9 Let (T, p) be a locally compact separable metric space, and let
{X(t), t£T}be an a-stable process satisfying Condition 10.6.4. If the process
{X{t), t G T) ispointwise sample continuous, then it is sample continuous.
PROOF- By pointwise sample continuity and Proposition 10.6.2 we conclude that
in Theorem 10.6.8, ax(t) = 0 for every teT.lt now follows from Proportion
9.3.5 that {X(t), t e T} is sample continuous. I
We have seen that an a-stable process {*(*), t € T} satisfying Condition
10.6.4 has a non-random oscillation function {ax(t), t € T}. What are possible
shapes for ax(-)? Recall that a function f^T - [-oo, oo] is said to be upper
semi-continuous if for every t€T, f(t) > lim3_t/(s)-

482 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.7
Property 10.6.10 The oscillation function {ax(t), t € T} is upper semi-
continuous.
The proof is left as an exercise.
Remarks
1. This is an important example of a situation where the knowledge of the
finite-dimensional distributions is clearly insufficient. If we are told that
the oscillation process {Wx(t), t G T} is 0 in the sense of the finite-
dimensional distributions,wzcmnotconcludzthat{X(t), t £ T} is sample
continuous because the SaS motion, 0 < a < 2, which is not sample
continuous, also has oscillations 0 (Example 10.6.3). In order to reach the
desired conclusion we would need to know additionally that the oscillation
process is non-random, and this fact can certainly not be determined by the
finite-dimensional distributions of the oscillation process.
2. Choosing different Tbs provides oscillation processes which are not only
versions of each other but also share some additional properties: they are all
non-random under Condition 10.6.4 (Theorem 10.6.8) or all random under
some other condition (Theorem 10.11.1).
3. The oscillation process {Wx(t), 0 < t < 1} of a Poisson process
{X(t), 0 < t < 1} is random and takes values in {0,1}; the value 1
at t = to indicates that X has a jump at t0.
4. We noted that the oscillation process of a (symmetric) stable motion is also
random. Now a stable process can be viewed as an integral with respect
to stable motion. Theorem 10.6.8 can then heuristically be understood as
follows: if the integrand is not uniformly continuous, then we may notice
the individidual jumps of the underlying stable motion and the resulting
oscillation may be expected to be random. Condition 10.6.4, therefore,
requires the integrand to be uniformly continuous. The condition is very
close to being necessary (Theorem 10.11.1).
5. The same heuristic argument explains why Condition 10.6.4 is also a
necessary condition for sample continuity (Theorem 10.3.1.)
10.7 The case 0 < a < 1
Property 10.7.1 Let {X(i), t € T} be ana-stable process, 0 < a < I, satisfying
Condition 10.6.4. Then its oscillation function {ax(t), t 6 T} takes values in
the two-point set {0, oo}.

10.8
THE CASE 1 < a < 2
483
PROOF: Assume, without loss of generality, that T](t) = 0 for all t &T. Then
{X(t), t € T} is strictly a-stable, and by Proposition 10.6.7, for every t € T,
ax(t) = Wx(t) = W2-./-(x1+JCa)W = 2-,/QWx1+Xl(t)
< 2-,/a(WXl(t) + WX2(t)) = 21-'/QQx(i) a.s.,
where {Xj(t), t €T}, i = 1,2, are independent copies of {X(t), t € T}. The
last relation implies ax(t) € {0, oo}. I
We are now able to describe fully the sample path behavior of a-stable
processes, 0 < a < 1, satisfying Condition 10.6.4.
Theorem 10.7.2 Let {X(t), t 6 T} be an a-stable process, 0 < a < 1,
satisfying Condition 10.6.4.
(i) There is an open set Tcont C T such that {X(t), t e T} is sample continuous
on TCont and is not sample bounded on any open set A such that AnT^ont ^ 0.
(ii) For every to € T, {X(t), t £ T} is either sample continuous at to, or is
not sample bounded in any neighborhood of to.
{Hi) If {X(t), t € T} is sample continuous at to € T, then it is sample
continuous in a neighborhood of to.
PROOF: (i) The set Tconl = {t e T: ax{t) = 0} is open by Property 10.6.10
and has the required properties.
(ii) The statement follows from (i) since either to 6 TcoM or to G T^OM.
(iii) Sample continuity at t0 implies to € Tcont. The claim follows because
TCont is open. I
10.8 The case 1 < a < 2
What happens in the case 1 < a < 2 ? Can the oscillation function take values
other than 0 or oo? The answer to the latter question is "yes," as the following
example demonstrates:
Example 10.8.1 Let 1 < a < 2, let VI, V2 ... be a sequence of i.i.d. SaS random
variables, and let Wx = VUW2 = -V,, W3 = 2~X{VX + V2), W4 = 2"'(VI -
V2),VV5 = 2-1(-V1+V2),W6=2-1(-Vi-V2),W7 = 3-1(Vi+V2 + V3),etc.
By the strong law of large numbers,
fim^ooWn = lim n-1 (|V,| + ... + |V„|) = B|V,| a.s.,
n—>oo v

484 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.9
Now let X(0) = 0, X(l/n) = Wn, n = 1,2,..., and define X{t) for all
other t e (0,1) by linear interpolation. The process {X(t), 0 < t < 1} is
SaS and satisfies Condition 10.6.4. However, ax{t) = 0 for t e (0,1] and
^x(0) = E\V\\ € (0,oo). For a similar construction in the case a = 1 see
Exercise 10.18.
The following properties of the oscillation function hold for all 0 < a < 2.
(They follow trivially from Property 10.7.1 in the case 0 < a < 1.)
Proposition 10.8.2 Let {X(t), t £ T} be a SaS process satisfying Condition
10.6.4. Then, for each ta € T,
p(l!nV.t„, igTo X{t) = X(t0) + -ajr(to),
lim^to, ter„ X{t) = X(to) - ^ax(to)) =.1. (10.8.1)
Proof: Let
Ux(tQ) = IS5t_t0. t6To (X(t) - X(t0)). (10.8.2)
Since {X(t), t €T} satisfies Condition 10.6.4, conclude that Ux(to) is, in fact,
a constant, i.e., there is a (3(to) S [0, oo] such that the event Qi = {u>: Ux {to) —
P(to)} occurs with probability 1 (see Exercise 10.19). By the symmetry of
{X(t), t 6 T}, we have Ux{t0) = Vx(*o), where
Vx(t0) = liiHt^, t€r„ (X(t0) - X(t)).
Therefore, the event Q2 = {w: Vx(to) = /3(<o)} occurs also with probability 1.
Let Oo be the null event of Theorem 10.6.8. For uefljnQifl Q.2, we have
ax(*o) = limt^—to, t,,heT0 \X(tx) - X(t2)\
= firn"tl^to, tleT0 (X{U) - X(t0)) +IiSt2_l0, t2€To (X{t0) - X{t2))
= 2jS(to),
i.e., P(to) = \ax{to)- This proves the result. I
10.9 The level sets of the oscillation function
Condition 10.6.4 neither implies nor is implied by continuity in probability
(Exercise 10.20). Nevertheless, the following result holds:
Proposition 10.9.1 Let {X{t), t € T} be an a-stable, a ± 1, or SIS process
satisfying Condition 10.6.4. If there is at0 e T such that ax (to) < 00, then there
is an r > 0 such that {X(t), t e J5(t0, r)} is continuous in probability.

10.9 THE LEVEL SETS OF THE OSCILLATION FUNCTION 485
PROOF: Let {*„, n > 1} C T be such that tn -> t0 as n -> oo. We shall prove
that -X(to) = plimn_^00X(tn). Let To be, as usual, a strong separant for X. We
can assume, without loss of generality, that {*„, n > 1} C To, since this will not
affect ax- The assumption ax(to) < oo implies
P( sup |Jf(i)| < oo) = 1
for some r > 0, and so
/ sup \f(t,x)\am{dx) <oo (10.9.1)
./£ t6T0nB(40,r)
by Theorem 10.2.3. Relations (10.6.1), (10.9.1), and the dominated convergence
theorem then yield
lim / |/(to, x) - f{tn, x)\am{dx) = 0.
n—°°JE
Applying Proposition 3.5.1, we conclude that X(to) = plimn_>0OX(in), i.e., the
process {X(t), t € T} is continuous in probability at to. Obviously, cxx{t) < oo
for every t € B(to,r), and the same argument shows that the process is continuous
in probability at every point of B(to, r). I
We have now the tools needed to prove
Property 10.9.2 Let {X{t), t £ T} be an a-stable, a ^ 1, or SIS process
satisfying Condition 10.6.4. Then for every e > 0, the set
TE = {tsT:ax(t)€[e,co)}
is nowhere dense.
Proof: Suppose firstly that the a-stable process {X(t), t € T} is symmetric,
and assume to the contrary that there is a non-empty open set A C T such that T£
is dense in A. Property 10.6.10 implies ax(t) > e for every t € A. Now take a
to € Te n A and observe that there is an r > 0 such that e < ax{t) < oo for all
t £ J3(io, r). Let Qq be the null set in Theorem 10.6.8 and
£2, = |w g Ag: X(t) = Hmu_t> u€To X(u) - -QX(«) Vt e T0 n B(t0, r),
X(t0) = Hmu_t0iUeTo^('")-2a^(t")}

486 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.10
By Proposition 10.8.2, P(Qi) = 1. Fixing an u e Clu we have
X(t0) = limu_t0iU6ToX(u) --a.v(io)
= limu_lo, u€r0 (lim,_u> seTo X(s) - -ax(u)j - -ax(to)
< --e + limu_t0) u6T0 Hms_Ui s€To X(s) - -ax(to)
< "J6 + lim«-t0, ueT0 A"(ix) - rdx(fo)
= -|e + X(to).
This contradiction proves our claim in the symmetric case.
In the general case a ^ 1, let {Xi(t), t € T} be two independent copies
of {*(*), t 6 T}, and let y(t) = Xx(t) - X2(t), t € T. Then {K(t), t 6
T} is a SqS process satisfying Condition 10.6.4. Let ax and ay denote the
corresponding oscillation functions. Suppose firstly ax (to) < oo for some *o €
T. Then {^(£), t € T} is continuous in probability in a neighborhood of i0
by Proposition 10.9.1. Proposition 10.6.7 (v) implies ay(t0) > ax{to). If
olx (*o) = oo, then ay (to) = oo as well (see the proof of Theorem 9.5.4). Using
Proposition 10.6.7 (iii), we conclude that, in all cases, ax(t) < ay(t) < 2ax(t)
for every t € T. Hence Te is a subset of the set Tey = {teT: aY(t) € [e, oo)}.
Since we have proved that T£y is nowhere dense, the set Te is also nowhere dense,
and the proof is now complete. I
Remark. We supposed that the process is symmetric in the case a = 1. The
above argument shows, however, that if the process is 1-stable, satisfies Condition
10.6.4 and is continuous in probability, then Property 10.9.2 holds as well.
10.10 A sample path alternative
Recall that the Belyaev's alternative states that the sample paths of a stationary
Gaussian process are either a.s. continuous or a.s. unbounded. We now state
a corresponding alternative for the sample paths of either a stationary a-stable
process or a self-similar a-stable process.
Theorem 10.10.1 Let{X(t), t e T) be an a-stable process satisfying Condition
10.6.4. If a = 1, assume additionally that {X(t), t £ T} is either symmetric or
continuous in probability.

10.10
A SAMPLE PATH ALTERNATIVE
487
(0 Suppose that (T,p) is a locally compact separable metric group1, and
{X(t), t £ T} is stationary. Then either {X{t), t £ T} is sample
continuous or {X(t), t £T} is not sample bounded in any non-empty open
subset ofT.
{ii) Suppose T — (0, co), and that {X(t), t £ (0, oo)} is H-self-similar (i.e.,
{X(ct), t £ (0,oo)} = {cHX(t) t £ (0,oo)} for every c >-0). Then the
conclusion of(i) holds as well.
Proof: (i) The stationarity of {X(t), t£T} implies that {ax(t), t € T} is a
constant function and it follows from Property 10.9.2 that either ctx(t) — 0 for
every t £ T, or ax(t) = oo for all t £ T. In the former case, {X(t), t £ T}
is sample continuous (Proposition 10.6.2 and Corollary 10.6.9), whereas in the
latter, {X(t), t £ T} cannot be sample bounded in an open ball around any given
point in T (Proposition 10.6.2).
(ii) It is easy to check that the a-stable process {Y(t) = e~HtX(et), t £
K} is stationary. But {X{t), t £ (0, oo)} is sample continuous if and only if
{Y(t), ieR} is, and there is an open set A in (0, co) such that {X(t), t £ A}is
sample bounded if and only if there is an open set A in M such that {Y(t), t € A}
is sample bounded. Part (i) now implies the result. I
Let us summarize our findings. We proved that any a-stable process
{X(t), t e T} satisfying Condition 10.6.4 has a non-random oscillation
function {ax{t), t e T} which is upper semi-continuous. If 0 < a < 1 then,
this function is, in fact, a function from T to the two-point set {0,oo}, whereas
if 1 < a < 2, the function must have nowhere dense level sets of the form
{t £ T: e < ax(t) < oo}. The same is true in the case a = 1 if {X{t), t € T}
is, additionally, either symmetric or continuous in probability.
It turns out, at least in the case T = Ed, that the oscillation function has, in
general, no additional properties, i.e., the following is true:
Proposition 10.10.2 Let d > 1 and let {a(t), t 6 Rd} be an upper semi-
continuous functionW.d —► [0,oo] such that for every e > 0 the set {t e M.d: a(t) 6
[e,oo)} is nowhere dense. Then there is a SaSprocess {X(t), t € Ed}, 1 < a <
2, continuous in probability and satisfying Condition 10.6.4, whose oscillation
function is precisely {a{t), t € Rd}. lf{a{t), t £ Rd} takes values in {0,oo},
then one can choose {X(t), t £ Rd} to be a SaS process with 0 < a < 1.
The proof of Proposition 10.10.2 follows closely the corresponding proofs in the
Gaussian case a = 2 given by Ito and Nisio (1968) and Jain and Kallianpur
'The parameter space T of a stationary process must be a group, e.g., T = R.

488 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.11
(1972). We explain in Exercise 10.23 how to adapt the Gaussian construction to
the stable case.
10.11 How strong is the basic assumption?
We have been assuming that Condition 10.6.4 holds. But must any integral
representation of an a-stable process {X(t), t € T} with non-random oscillation
satisfy that condition? Of course, the requirement that {??(£), t € T} be
continuous is not necessary for {X(t), t € T} to have a non-random oscillation. What
about the second part of Condition 10.6.4, namely (10.6.1)?
Observe that the well-balanced symmetric linear fractional stable motion
(Example 3.6.5) with H < 1/q is not sample bounded on any interval of positive
length (see Example 10.2.5) and thus its oscillation process {Wx(t), t e R} is
non-random; in fact, ax (t) = oo for every t 6 R. Since, clearly, the
representation (3.6.4) of the well-balanced symmetric linear fractional stable motion does
not satisfy Condition 10.6.4 when H < 1/q, we have a counterexample.
It turns out that the above counterexample points to the only way that we can
have a non-random oscillation when the integral representation (10.1.1) does not
satisfy the second part of Condition 10.6.4: this non-random oscillation must be
infinite. More precisely, the following holds:
Theorem 10.11.1 Let {X(t), t € T} be an a-stable process given in the form
(10.1.1). If a = 1, suppose, additionally, that {X(t), t € T} is continuous
in probability. Suppose that the oscillation process {Wx(t), t e T} is non-
random and finite, i.e., there is an event Qq € £1 with P{Qq) = 0 and a function
aX-T -+ R+ such that for every ui € Qg, every t 6 T, Wx(t) = ax(t). Then
for every countable dense subset T* ofT, every compact subset C ofT,
m{x: the function f{-,x): T* D C —> R is not uniformly continuous) = 0.
Remark. The conclusion of Theorem 10.11.1 holds if {X(t), teT} is SIS; in
this case, one does not need to assume that X is continuous in probability because
continuity in probability follows from the argument in the proof of Proposition
10.9.1.
PROOF OF THEOREM 10.11.1: The assumptions of the theorem and the argument
in the proof of Proposition 10.9.1 imply that {X{t), t € T} is continuous in
probability. Let T* be a countable dense subset of T and let C be a compact
subset of T. Since C is compact,
WX(C) = sup ax {t) =: ax(C) < oo a.s.

10.11
HOW STRONG IS THE BASIC ASSUMPTION?
489
as ax is, obviously, upper semi-continuous. In particular, Wx(C) is non-random.
Let {Xi(t), t€T},i= 1,2, be two independent copies of {X(t), t € T}, and
lety(t) = Xi(t)-X2(t), teT. Then{y(i), t € T} is a SaS process with an
integral representation
Y(t) = 21/Q / f(t,x)M(dx), t € T,
where M is a 5a5 random measure on (E, S) with control measure m. Note that
by Proposition 10.6.7,
WY(C)<2ax(C) a.s.
and, in particular, {Y(t), t € C} is sample bounded. Applying Theorem 10.2.3
we obtain
/ f*(T*nC-x)am{dx) < oo,
where f*(T* flC;i) = supt€T.nC |/(t,i)|. Therefore {Y(t), teT*nC} has
the following integral representation:
Y(t) = / , J?'*! x M*(dx)-, ternc,
7e+ /*(T*nC;x) v
where E+ - {x e E : f*{T* f\C\x) ± 0} and M* is a SoS random measure on
(E,S) with a^«ifc control measure m*(dx) = 2/*(T*nC;x)am(dx). Applying
Theorem 3.10.1, we obtain {Y(t), i 6 T* n C} = {Z{t), t e T* n C), where,
in the notation of that theorem,
Z(t) = (CQm*(B))|/"X:^rr1/Q7^S^)' * 6 T* nC'
/(*,Vi)
i=i ^
In particular,
nSd(t„t2)-,o, tl.t,€T-nc l^(*i) - Z(i2)| < 2ax(C) a.s. (10.11.1)
We now use a "trick" similar to the one we applied in the proof of Theorem 10.2.3.
Let?! =€[,?; = —£j, i > 2, and define
-l/q /(*.y<)
= 1
Then {Z(t), t£T*nC} = {Z(t), i € T* n C}. If I/(t) = Z{t) + Z(t), t 6
T'nC, then by (10.11.1),
rr1/ar(T*nC;V'1)-1iirH(f{t,,e2)_>o,t,,t2€T-ncl/(ii.^)-/(*2^i)l
= nrHd((l,t2)_o, t„t2€T-nc {Cam'{E))-l'aMU) - Wl
<4(CQm*(£))-1/oax(C)a.s. (10.11.2)
Z(t) = (CQm*(E))'/^?1rr1/a77^p^y, t e T* nc.

490 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.12
This implies
m*(x€E: firn^^o, t.^er-nc \f(t\,x) - f(t2,x)\ ^ 0) = 0. (10.11.3)
Indeed, suppose that (10.11.3) does not hold. For any V\ belonging to the set on
the left-hand side of (10.11.3), there is a positive probability that the left-hand
side of (10.11.2) exceeds the constant on its right-hand side (this happens when
T\ is small), and thus, by Fubini's theorem, (10.11.2) is violated.
Observe lastly that (10.11.3) is equivalent to the statement of the theorem. I
10.12 Exercises
Exercise 10.1 Prove that fractional Brownian motion is sample continuous.
Hint: Prove that E{X(t) - X{s))ln = a2n\t - s\2Hn, n > 1, and use Kol-
mogorov's criterion for sample continuity.
Exercise 10.2 Letu0 > 0 and define um = 21/Qum_i +u^_,, m > 1, for some
a > 0 and 6 G (0,1). Show that the limit / = limm_00 2~m/aum exists and is
finite.
Hint: Use 0 < 2~1/aui < 2_2/au2 < • • • to show that the limit exists and is
non-negative. To prove that it is finite show that for k > 1 and m satisfying
2-m/aum > 1,
k
2-(m+fc)um+fc < 2-mlaumY[U+2~j^~e)la)
i=i
OO
< 2-m/QumJ]exp{2-^1-e)/Q}.
j=i
Exercise 10.3 Let an G (0,1), n > 1. Show that fl^Li an > 0 if and only if
£r=i(l-«<oo.
Exercise 10.4 (i) Let (E, £, m) be a cr-finite measure space. Construct a
probability measure P on (E, £) equivalent to the measure m.
(ii) Give an example of a measurable space (E, 8) on which there are two
equivalent measures, m and P, where P is a probability measure and m is not
CT-finite.
Exercise 10.5 Prove Corollary 10.3.2.
Exercise 10.6 Prove Theorem 10.3.3.
Hint: You may find it useful to use the zero-one law of Theorem 9.5.4 (i) to prove
that {X(t), t G T} is sample bounded in a neighborhood of to.

10.12
EXERCISES
491
Exercise 10.7 (i) Let 1 < a < 2, {an, n> 1} be a sequence of positive numbers
satisfying (10.4.1) and (10.4.2) and let bn = an{n(n + 1))'/°. Let
X{t) = J f{t,x)M(dx), 0 < t < 1,
Jo
where M is a SaS random measure on ((0,1), B) with Lebesgue control measure
and define the functions {f(t, x), x G (0,1)}, 0 < t < 1, as follows:
Let t = 0.d\d2 ... be the binary expansion of a t € (0,1). For N — 1,2,...,
define
Tjy = |t € (0,1): t = 0.did2 ..., di = 0 Vi < N - 1,
where d/v = l,d2W+i = l,<ii = 0 Vi>2AT + 2J.
For at £TN let A{t) = {i E {N + 1,... ,2N}: di = 1} and set
' ^n ifx 6(^,1] and n + iVeA(i),
/(t,i)=< -6„ ifx€ (^t,^] and n<N but n + iV 0 y4(t),
0 otherwise.
To complete the definition of /, set /(0, x) = f(l,x) = 0 Vx e (0,1) and define
f{t,x) for the remaining ts in (0,1) by interpolating linearly between the is in
Un=.^u{o,i}.
Show that {X(t), 0 < t < 1} is a well-defined SaS process. Show also that
the kernel /(£, x) is strongly separable, that
/ sup \f(t,x)\adx < oo,
Jo o<t<i
but {X(t), 0 < t < 1} is not sample bounded.
Hint: For the last question, define Zn = {{n(n + l))1/QM((^rr, i])} and show,
as in the text, that J^Li an|Zn| = oo a.s.
(ii) Of course, the SaS process in (i) is also not sample continuous either,
but this is not a good counterexample for sample continuity because the functions
{f(t,x), t 6 [0,1]} are not uniformly continuous on rational numbers for any
x e (0,1). We can easily overcome this point. Let Zn be defined as above and
let Sn = Yln=i an\Zn\- You showed in (i) that Sn T Qo a.s. as N -> oo.
Show firstly that there is a non-decreasing sequence of integers {kn, n =
1,2,...} such that fcn | oo as n —► oo, and limiv-»oo «5jv - SkN — oo a.s.

492 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.12
For t € TV, redefine f(t, x) as follows:
bn if a; € (^y, £] and kN < n < N and n + N € A(t),
f(t,x)=l -bn ifx€ (^,^]andkN<n<N{0Tn + N ^A(t),
0 otherwise,
and, as before, define f(t, x) for the rest of the is by linear interpolation.
Show that the resulting kernel f(t, x) is strongly separable, that {f(t, x), 0 <
t < 1} is continuous for every x € (0,1) and that
I
l
sup \f(t,x)\adx < oo.
0 t€[0,l]
Show that {X(t), 0 < t < 1} is not a.s. bounded and therefore not sample
continuous.
Exercise 10.8 Give a direct argument showing that (10.4.10) implies (10.4.5).
Hint: Define a new a-stable process W{(t,d)) = {-l)dX(t), t e T, d € {0,1}.
Exercise 10.9 Prove the second part of Corollary 10.4.3.
Exercise 10.10 Let / be a measurable function R —» M satisfying
Leb(x e R: /(r + x), r € Q is not uniformly continuous) = 0,
where Q denotes the set of rationals.
(i) Show that for any xq € R there cannot exist two numbers a > b such that
for every 6 > 0,
Leb(x e R: f{x) > a, \x - x0\ <6)>0,
Leb(x € R: f(x) < b, \x - x0\ < 8) > 0.
Hint: You may find it useful to prove first that for any two measurable sets A, B
with Leb(A) > 0, Leb(J5) > 0 there are points x 6 A and y € B such that
x - y G Q.
(ii) For an io € R, define
a(x0) = sup ja e R: V<5 > 0, Leb(x: f(x) > a, \x - x0| < 6) > o|,
b(x0) = inf{6 € R: V<5 > 0, Leb(x: /(x) < 6, |x - x0| < <5) > o}.
Show that a(xo) = b(xo).

10.12
EXERCISES
493
(iii) Show that the function
g(x) = a(x) (= b(x)), x € K,
is continuous.
(iv) Prove that Leb (x: g{x) ^ f(x)) = 0.
Exercise 10.11 Prove Theorem 10.4.6.
Exercise 10.12 Let W\, W2,... be i.i.d. random variables such that
lim XaP{Wl >X) = H
A—>oo
for some a > 0. Let N be a Poisson random variable with mean m, independent
of the sequence W\, W2, ■ ■ ■. Show that
N
lin^ AQP(J] W; > A) = mif.
Exercise 10.13 Let (T, d) be a separable metric space and let T0 and T] be the
respective strong separants of two strongly separable versions of a stochastic
process {X(t), t 6 T}. Set
Wi(t) = iimtl,t1-,t. t„t1€T1 |X(tz) - *(*i)|, ter, i = 0,1.
Show that {W0{t), t&T} = {Wx(t), t € T}.
Exercise 10.14 Let (E,£,m) be a measure space and suppose that there is a
never-vanishing function g : E —> E such that g e La (m) for some a > 0. Show
that to must be er-finite.
Hint: Let B0 = {x : \g{x)\ > 1}, Bn = {x : (n + l)"1 < |ff(a;)| < n"1}, n =
1,2,..., and note that m(Bn) < 00 for each n.
Exercise 10.15 Prove the rest of Proposition 10.6.7.
Exercise 10.16 Prove Relation (10.6.5).
Exercise 10.17 Prove Property 10.6.10 of the oscillation function.
Exercise 10.18 Let V,, V2,... be a sequence of i.i.d. 51S (Cauchy) random
variables. Use the weak law of large numbers (Theorem VII.8 II of Feller (1971)) to
conclude that there are numbers {an, n > 1} such that lim^ooU"1 £"=1 |Vj| =
1 a.s. Use this to construct, as in Example 10.8.1, a 515 process {X(t), 0 < t <
1} satisfying Condition 10.6.4 such that its oscillation function {a{t), 0 < t < 1}
takes values outside the set {0,00}.

494
BOUNDEDNESS, CONTINUITY AND OSCILLATIONS
10.12
Exercise 10.19 Let {X(t), t € T} be an a-stable process satisfying Condition
10.6.4, and let {Ux(t), t e T} be defined by (10.8.2). Show that there is a
function /?: T -+ [0, oo] such that for every t € T, P(Ux(t) = 0(t)) = 1.
Exercise 10.20 (i) Construct an example of a SaS process satisfying Condition
10.6.4 but not continuous in probability.
(ii) Construct an example of a SaS process continuous in probability which
does not satisfy Condition 10.6.4.
Hint for (i): Let T = [0,1] and m = Leb. Let /(£, x) = n2/Q l,_j_ u (x), n =
1,2,..., /(0, x) = 0, and define the rest of /(£, x) by linear interpolation in t.
Hint for (ii): The SaS L6vy motion.
Exercise 10.21 Let d > 2, and let M be a SaS random measure on (Rd,Bd)
with d-dimensional Lebesgue control measure. Let 0 < H < 1, and define for
X{t)= f (||x-tl||"-d/a-||x||H-d/Q)M(dx),
wherex = (x1,...,xd),l = (l,...,l)and||x||==(EtIx?)1/2.
(i) Show that {X(t), t £ 1} is a well-defined SaS process, continuous in
probability and satisfying Condition 10.6.4.
(ii) Show that {X{t), t e R} is if-self-similar and has stationary increments.
(iii) Show that {X(t), t € M.} is sample continuous (and hence bounded on
bounded intervals) if H > I/a, a > 1. Recall that it is not sample bounded on
any interval of positive length if H < \/a (Example 10.2.5).
Exercise 10.22 Modify the construction of Example 10.8.1 to construct a SaS
process {X(t), 0 < t < 1}, 1 < a < 2, satisfying Condition 10.6.4 and having
the following property: for given 0<u<u<l,a>0,
(i) X(t) = 0 a.s., 0 < t < u,
(ii) {X(t), u < t < v} is sample continuous,
(iii) limtiuX(t) = a a.s.,
(iv) X(i) = X(v) a.s. for all t £ [v, 1].
Exercise 10.23 Follow the proof of Theorem 4(b) of Ito and Nisio (1968) in order
to construct a SaS process {X(t), 0 < t < 1} with a given oscillation function.
Go through the following steps:
(i) Given 0<a<2, 0<u<u<lande>0 construct a SaS process
{X(t), 0 < t < 1} satisfying Condition 10.6.4, continuous in probability, and
such that
(a) X(t) = 0 a.s. for every t £ (u, v),
(b) ax(t) — co for every t € [u, v),

10.12
liXERCISES
495
(c) <?x(t) < (■ for every t G [0,1].
Here ax{t) is the scale parameter of the SaS random variable X{t).
Hint: Use Exercise 10.21 and follow the proof of Lemma 6.1 of ItO and Nisio
(1968).
(ii) Given 1 < a < 2, a > 0, e > 0 and 0 < u < v < 1, construct a SaS
process {X(t), 0 < t < 1} satisfying Condition 10.6.4, continuous in probability
and such that
(a) {X(t), 0 < t < 1} is sample continuous,
(b)X{t) = 0a.s. if* g(u,v),
(c) <7x(t) < e for every £ € [0, I],
(d)P(|suPl,<t<„X(t)-a| >«)<«•
//i/ir: Use Exercise 10.22 and follow the proof of Lemma 6.2 of Ito and Nisio
(1968).
(iii) Let{Qi(i), 0 < t < 1} be two upper semi-continuous functions such that
thesets{i e [0,1]: <*(*) > 0},i = 1,2, are nowhere dense. Let{X,(t), 0 < t <
1} be a SaS process, 1 < a < 2, continuous in probability, satisfying Condition
10.6.4, such that ax>(t) = a,(t) for all t € [0,1]. Show that for every e > 0
one can construct a SaS process {X2(t): 0<t<l}, continuous in probability,
satisfying Condition 10.6.4 and independent of the process {X,(t), 0 < t < 1},
such that
(a) aXl+Xl(t) = a,(t) + a2(t) for every t 6 [0,1],
(b) ox,m < e for every * e [°> 11'
(c) for "separable version {Y2(t): 0 < t < 1} of {X2(i), 0 < t < 1} we
have s
p( sup \X2(t)\> sup a2(i)J <£•
V[o,i] teto.il
Hint: Follow the proof of Lemma 6.3 of Ito and Nisio (1968) In the case a = 1
you will need a slightly different argument to show that {X2{t), 0 $t £ 1} «
continuous in probability, and you will have to show additionally that {X2(t), 0 <
t < 1} satisfies Condition 10.6.4.
(iv) Let {a(*), 0 < i < 1} be an upper semi-continuous function [0,1] -
[O.oo] such that! for any e > 0, the set {t € [0,1]: « < «(*) < oo -nowhere
dense. Prove that there is a SaS process {X(t), 0 < t < 1}, a < ,
continous in probability, satisfying Condition 10.6.4 and such that ax(t) = a{t)
for all t e [0,1].
//mr: Follow the proof of Theorem 4Toj of It6 and Nisio (1968)
(v) Let {«(*), 0 < * < 1} be an upper semi-continuous function [M -
{0,oo} and let 0 < a < 1. Construct a SaS process {X{t), 0 * t < 1}
continuous probability, satisfying Coofition 10.6.4 and such that ax(t) - a{t)
for every t e [0,1].

496 BOUNDEDNESS, CONTINUITY AND OSCILLATIONS 10.12
Hint: The set T0 = {t £ [0,1]: a(t) = 0} is open and hence is a countable union
of open intervals. Write T0 = tXLi(an>&n)> where (a„,6n) are disjoint open
intervals. For n = 1,2,..., define
X(t) = J " f(t,x)M(dx), an<t< bn,
Ja„
in such a way that {X(t), an <t < bn) is sample continuous, limt_an \X{t)\ =
limt^b„ \X(t)\ = oo, crX(t) -* 0 as t -+ an, t -> 6n, and for any x G (an, 6„),
limt_0„ /(i, x) = limt^6n f(t, x) = 0.
Then define X(t) on the closed set T^ = {t € (0,1): a(t) = oo} as follows:
if t is a boundary point of Too, set X(t) = 0 a.s.. If / = [a, 6] is a maximal
subinterval of Too, then define on (a, 6)
X(t) = I f{t,x)M(dx), a<t<b,
Ja
in such a way that {X(t), t € (a, b)} is nowhere bounded, continuous in
probability, limt_*a aX(t) — limt_6 cx(t) = 0, and for every x € (a, 6)
lim /(£, x) = lim /(*, x) = 0.
t—ta t—*b
(vi) Generalize the above construction to get SaS processes on [0, l]d with a
given oscillation function.
Hint: Follow the proof of Theorem 4 of Jain and Kallianpur (1972).
(vii) Prove Proposition 10.10.2.

Chapter 11
Measurability, integrability
and absolute continuity
In this chapter, we consider the integrability of a-stable processes. For a process
to be integrable, it must be measurable, and we have already seen in Chapter 9
that not all stochastic processes have a measurable version. We give in Section
11.1 necessary and sufficient conditions for measurability in terms of the integral
representation of the process.
Suppose that the a-stable process is given as {X{i) = fBf(t,x)M{dx),
t 6 T}. In Section 11.2 we consider the integrability of that process with respect
to t. More precisely, we investigate whether JP(X) = JT \X(t)\pi>(dt) converges
or diverges with probability 1, for a given p > 0 and cr-finite measure v. Necessary
and sufficient conditions for convergence are given in Section 11.3.
In Section 11.4 we determine conditions under which we can change the order
of integration. Theorem 11.4.1 states thatif J \X(t)\u(dt) < ooa.s., then the order
of integration can be changed, i.e., JT X{t)v(dt) = JE(JT f(t, x)v{dt))M{dx).
In Section 11.5 we investigate properties of the random "Lp-norm"
(Jp(X))'/p = {JT |X(i)|pi/(dt))'/p. The random variable (Jp(X))1/p is a norm
on the sample paths of {X{t), t € T} when p > 1. Theorem 11.5.1 specifies the
asymptotic behavior of P{(Jp(X))1/p > A}, p > 0, as A -> oo if JP{X) < oo
a.s.
This last result is used in Section 11.6 to obtain an inversion formula for
the harmonizable SaS process Z(t) - ^ooeitxM{dx), -oo < t < oo. The
process Z can be viewed as a Fourier transform of the random measure M.
Theorem 11.6.1 provides an inversion formula that expresses M in terms of Z.
In Section 11.7 we consider absolute continuity. Absolute continuity implies
the existence of a derivative. Let ACp[a,b] denote the set of all absolute con-

498 MEASURABILITY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.1
tinuous functions <j> on the interval [o, b] such that the derivative <j> of <j> satisfies
fa \4>(t)\Pdt < oo, p > 1. Theorem 11.7.4 gives necessary and sufficient
conditions for an Q-stable process, X(t) - fEf(t,x)M(dx) + r](t), t G [a,b], to
have an absolute continuous version in Lp[a,b], p > 1. The conditions involve
the functions / and 77.
11.1 Existence of a measurable version
Let {X(t), t e T} be an a-stable process on a separable metric space (T, p).
How can we find out whether the process admits a measurable version? The
following theorem gives a necessary and sufficient condition. It has a number of
important corollaries which will be stated below.
Theorem 11.1.1 Let {X{t), teT}be an a-stable process, 0 <C ot < 2, with an
integral representation (10.1.1). The process X has a measurable version if and
only if the following conditions hold:
(a) There is a (jointly) measurable function g: T x E —> R such that for every
teT
m(x e E: f(t,x) ^ g{t,x)) = 0
(i.e., the kernel f(t, x), t € T, x £ E has a measurable modification.
(b) There is a set Eq e £ such that the restriction of the control measure m to
Eg is a-finite and m(£0 f"l {x € E: f(t, x) ^ 0}) = Ofor every t e T (i.e.,
the measure m is essentially a-finite).
(c) The function tj:T —> K. in (10.1.1) is measurable.
Moreover, if {X(t), t € T} has a measurable version, thenthe linear subspace
of'La{E,S,m) spanned bythe functions {/(t, •), t € T} is necessarily separable.
Proof: Suppose firstly that conditions (a)-(c) of Theorem 11.1.1 hold. Let n be
a probability measure on (E, £) equivalent to the measure m restricted to E% and
set
Proposition 3.5.5 and Theorem 3.10.1 imply
00
{X(t), teT}± {C^(7ir71/Qff(i, Vj)r{Vj) ~ °i(«.*)) + »?(*), * € t},
(11.1.1)

11.1 EXISTENCE OF A MEASURABLE VERSION 499
where C > 0 is a constant, and, as usual, {(Vj), j > 1} and {Tj, j > 1} are
independent sequences: the first consists of i.i.d. random elements in fix {—1,1},
and the second of arrival times of a Poisson process of unit rate; Vj has distribution
n on E, P(7j = l\Vj) = 1 - P(7i = -1|V$) = (1 + /3(^))/2 and
' 0,
a,j(a,t) = <
Utiri) u'2 sinudu - 2_1/j - ln l) Se9(t,x)P(x)m(dx),
C.-I ,
sW^ - U - l)-^)JE9(t,x)P(x)m(dx)
(11.1.2)
in the cases 0<a<l,a=l and 1 < a < 2, respectively. Recall that the series
on the right-hand side of (11.1.1) converges with probability 1 for every t e T.
Forfc = 1,2,..., define
k
xk{t) = c^T^rj^g^VjyiVj) -aj{a,t)) + v(t), t e t,
and observe that the functions a.j(a, ■): T —> K are measurable under Condition
(a) of the theorem. Therefore the processes {Xk(t), t £ T}, k = 1,2,..., are
all measurable. Let
. . _ J limfc_oo^fc(£,w) if the limit exists,
otherwise.
Then {V(i), t S T} is a measurable process (since it is the pointwise limit of
measurable processes) and it is a version of {X(t), t e T} by virtue of (11.1.1).
We now turn to the converse and suppose that {X (i), i € T} has a measurable
version. Let {X(t), t e T] be an independent copy of {X(t), t S T} and let
Z[t) = 2-'/Q(X(i) - X(i)), t € T. Then {Z(i), i € T} is a SaS process that
has a measurable version and the representation
{Z(t), t£T}±{J f(t,x)M(dx), teT},
where M is a SaS random measure on {E, £) with the same control measure
m as M. Theorem 9.4.2 implies that {Z(t), t e T} satisfies Condition S and it
follows from Proposition 3.5.1 that the linear subspace of La{E,£,m) spanned
by the functions {/(i,-), t € T} is separable. Therefore the set {/(*,-), * e
T} C La(E,£,m) is separable as well. Let T* be a countable subset of T
such that for every t € T there is a sequence {in, n = 1,2,...} C T* such
that /(*„,-) -► /(*,-) in La{E,£,m) as n -► oo. It is easy to verify that
£0 = {z € E: f{t, x)^0 for some i € T*}c satisfies Condition (b) of Theorem
11.1.1 and
{Z(i), t € T} ^ {y* a f(t,x)M(dx), t e T}.

500 MEASURABIL1TY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.1
Let n(dx) and r(x) be defined as before. Theorem 3.10.1 implies that the SaS
process
oo
W(t) = J2ejTJl/af(t, Vj)r(Vj), t e T,
is (up to a multiplicative factor) a version of { Z(t), t e T} and therefore possesses
a measurable version. Here, {tj, j > 1}, {Tj, j > 1} and {Vj, j > 1} are
independent sequences of, respectively, Rademacher random variables, arrival
times of a unit rate Poisson process and i.i.d. random elements in E with common
law n. Of course, the choice of the space on which these three sequences are
defined is arbitrary. Let us choose this space to be (Q.\ x Q2 x Q3, F\ x^jx
Tz,P\ x P2 x P3), such that {eJt j > 1} is defined on (Q,, Fu P,), to, j > 1}
is defined on (Q^^Pz) and {V}, j > 1} is defined on (Q3,.T^Pj). More
specifically, let (£23, .F3, P3) = {E, £, n)N so that each Vj is defined on a separate
copy of (E, £,n), and let Vj (x) = x, x e E.
Now define e} = —€j, j > 2, ?j = e\, and let
00
w(«) = 2?iI7,/a/(t.^)»-(yi-), * e r.
Clearly, {^(i), t € T} is a version of {W(t), t e T} and, as such, has a
measurable version. But, by Theorem 9.4.2,
2e!r71/Q/(tIv1)r(y1) = w(t) + w(t))teT,
has a measurable version as well. It then follows from Theorem 9.4.2 that there
is a (jointly) measurable stochastic process {U(t,u\,uj2,x), t € T} defined on
the probability space (Qi x Q.2 x E, T\ x T2 x £, P\ x P2 x n) such that for any
teT
P,xP2xn{(wi,o;2,2;):2ei(wi)r1(w2)""1/o'r(x)/(i,x)^t/(i,w1,W2,a;)} = 0.
In particular, for any t 6 T and n-almost every x € E,
P, x P2{(u;ilW2): 2e1(w1)ri(w2)-,/Qr(i)/(t)x) ^ {7(t,w,,u;2,x)} = 0-
(11.1.3)
Define
./nl./G22ei(w,)ri(w2) '/«r(a:)
and g(i, x) = 0 otherwise. Then y(t, x) is a O'ointly) measurable function such
that for every t€T
m(x 6 £0C: /(t,x) ± g(t,x)) = n(x € .Eg: /(t,i) ^ <?(*,*)) = 0

11.1
EXISTENCE OF A MEASURABLE VERSION
501
and
m(x e E0: f{t, x) ^ g{t, x)) <m(xe E0: f(t, x) ^ 0) = 0.
Thus g{t,x) satisfies Condition (1) ofTheorem 11.1.1.
It follows from the already established sufficiency part of the theorem that
the a-stable process {X{t) = X{t) - rj(t), t € T} has a measurable version.
This means that {r){t) = X (i) - X(t), t € T} has a measurable version and,
hence, must be measurable (see Exercise 9.14). The proof of the theorem is now
complete. I
Remark. As the preceding proof shows, the sufficiency part of the theorem
continues to hold when the parameter space (T, T) is any measurable space (not
necessarily Borel).
Theorem 11.1.1 has a number of important consequences:
Corollary 11.1.2 A strictly a-stable, a ^ 1 or SIS, process {X(t), t € T} has
a measurable version if and only if it admits an integral representation of the form
{X{t), teT}~U f(t,x)M{dx), t 6 T}, (11.1.4)
where M is a totally skewed to the right a-stable random measure with Lebesgue
control measure and where f is jointly measurable and in La for each t.
This corollary is an immediate consequence of Theorem 11.1.1 and the general
representation theorem for strictly stable processes in Theorem 13.2.1 given in the
chapter on integral representation.
The next corollary follows easily from the proof of Theorem 11.1.1.
Corollary 11.1.3 Let {X{t), t e T} be an a-stable process, 0 < a < 2,
with an integral representation (10.1.1), and suppose that conditions (a)-(c) of
Theorem 11.1.1 are satisfied. Then {X(t), t 6 T} has the measurable version
{Y{t), t £ T), where for t 6 T,
oo
Y(t) = CxJa 5>jI71/aff(*. Vj)r{Vi) ~ «>>*)) + */(*)
i=i
if the series converges and Y (t) = 0 if the series diverges. Here,
n is a probability measure on E equivalent to the measure m restricted to
Eq, {(Vj)i 3 ^ 1} md iTh J ^ *} are independent sequences of,
respectively, i.i.d. random vectors in E x {-1,1} and arrival times of a unit rate

502 MEASURABILITY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.2
Poisson process, Vj has law n,
p(7i = i\vj) = i - p(7i = -iw) = 1 + P2{Vj\
and Ca and a.j(a, t) are given, respectively, by (1.2.9) and (11.1.2).
Remark. We have discussed measurability of real-valued a-stable processes. The
results apply also to the complex-valued case after some obvious modifications.
Example 11.1.4 By applying Theorem 11.1.1, we conclude immediately that
the SaS LeVy motion (Example 3.6.1), any SaS moving average (Example
3.6.2) , the well-balanced SaS linear fractional stable motion (Example 3.6.5)
and the log-fractional SaS stable motion (Example 3.6.6) all have measurable
versions. (This is true as well for their corresponding skewed analogs.) The
harmonizable SaS process of (6.5.1) and its real counterpart (Example 6.2.5) also
have measurable versions. The sub-Gaussian and sub-stable SaS processes have
measurable versions if and only if the underlying Gaussian or stable processes do.
11.2 Integrability of the sample paths of stable
processes
Let {X(t)t t € T} be a measurable a-stable process on a measurable space
(T, T), let v be a non-negative c-finite measure on (T, T) and let p be a positive
number. We want to determine whether or not the sample path integral
JP(X) = ^\X(t)\'v(dt)
converges.
We start with two observations. Firstly, because {X(t), t € T} is measurable,
Fubini's theorem implies that JP(X) is a well-defined random variable (taking
possibly the value +oo). Secondly, as in the case of sample boundedness and
sample continuity, there is a zero-one law for sample integrability.
Theorem 11.2.1 Let {X(i), t € T} be a measurable a-stable process on a
measurable space (T,T), let v be a o-finite measure on (T,T) and let p be a
positive number. Then
P(u: f \X(t,uj)\pv(dt) <oo) =0orl,
and the left-hand side depends only on the finite-dimensional distributions of
{X(t), t 6 T}.

11.2 INTEGRABILITY OF THE SAMPLE PATHS OF STABLE PROCESSES 503
Proof: Let (Q, T, P) be the probability space on which the process X —
{X(t), t G T} is defined, and let M be the family of all measurable stochastic
processes indexed by T defined on the probability space (O, T, P) =.(^> F, P)
and depending only on finitely many coordinates of the product space. Say that a
measurable function / on T has the property V if / € LP(T,T, v). Now apply
Theorem 9.5.4. Of all its conditions, the only one that cannot be trivially verified
is Condition (v). To show that it is satisfied, one needs only to prove that for
X e M, the probability P(Q: JT \X (t,Q)\pu(dt) < oo) depends only on the
finite-dimensional distributions of X. (Recall that X e M implies that X is
defined on (&,£,?).)
It is obviously sufficient to consider the case where v is a probability measure.
Let T\, r2,... be a sequence of i.i.d. T-valued random variables with common law
v defined on a different probability space (Q,,T\,P\). Then JT \X{t)\pv{dt) =
Ex\X{tx)\p. Exercise 1.24implies
Pi(ux 6 Q,: lim n-^pX(Tn(cjl),Q) = 0) =1 (11.2.1)
for any Q € A, where A = {Q € f2: fT \X(t,Q)\Pu(dt) < oo}. Therefore, by
Fubini's theorem,
P{A) = Py.Px{{Q,ux):Q€ A, lim nll*X[jn{tjj{),u)) = o).
Exercise 1.24 and Fubini's theorem also imply
PxPx ((C.wi): Q&A, lim n-'^XKfwO.cD) = o) = 0.
Therefore
P{A) = P x P, f(C.wi): lim n-1/pX(rn(wi),w) = o)
= £,fp(uGQ: lim n-,/pX(TB(u;I),tD) = 0)l.
Now, for every fixed ui\ € Qi, the probability
p(Qeh: lim n-1/pX(Tn(w,),w)=o)
depends only on the finite-dimensional distributions of {X(t), t&T) and hence
the same is true for P(A). Therefore Theorem 9.5.4 applies and the proof is now
complete. 1
We have just seen that the finite-dimensional distributions of a
measurable a-stable stochastic process {X(t), t € T} determine the probability
P{JT \X(t)\pv(dt)< oo). In fact, if that probability equals 1, they determine the
distribution of the random variable / \X{t)\pv{dt) (see Exercise 11.3).

504 MEASURABILITY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.3
11.3 Conditions for integrability
We have seen that the sample path integral is either finite a.s. or infinite a.s. Under
what conditions is it a.s. finite?
Let {X(t),t e T} be a measurable a-stable process with an integral
representation
X(t) = f f{t,x)M{dx)+r](t), t e T,
JE
i.e., it is given in the standard form (10.1.1), and assume that (T, p) is a separable
metric space. We may and will suppose without loss of generality that the kernel
f:TxE->R itself is jointly measurable and that conditions (b) and (c) of
Theorem 11.1.1 are satisfied. Let v be a non-negative cr-finite measure on (T, T)
where T is the Borel a-algebra on T generated by the metric p. We shall make
these assumptions throughout the remainder of this chapter unless stated otherwise.
The following proposition gives necessary integrability conditions on / for
{X(t), t 6 T} to have integrable sample paths.
Proposition 11.3.1 Let {X(t), t £ T} be a measurable a-stable process, 0
a < 2, with integral representation (10.1.1) and letp > 0. //
<
then
f \X(t)\"v(dt) <ooa.s.,
,p/'
J (J \f(t,x)\am(dx)\ u{dt) <oo (11.3.1)
and
a/p
/ (/ l/(t.aOIMd*)J m(dx) <oo. (11.3.2)
PROOF: We may assume without loss of generality that both m and v are
probability measures. Let (Q,.F,P) be the probability space on which the
process {X(t), t £ T} is defined, and let tut2,.-. be a sequence of i.i.d. T-
valued random variables with common law v defined on a different probability
space (Oi.-Fi.Pj). Then for P-almost every u £ Q, £'i|X(ti,w)|p < oo
and thus Exercise 1.24 implies that limn_00n-l/pX(Tn,a;) = 0 Pi-a.s., so
that, by Fubini's theorem, for P-almost every choice of tx , r2,..., we have
P{limn_00n_l/pX(Tn) = 0} = 1. Appealing to Theorem 10.3.1, we conclude
that for P\ -almost every choice of T\, T2,...,
/ sup I n ,/p|/(Tn,x)|) m(dx)<oo. (11.3.3)
J£n>P *

11.3
CONDITIONS FOR INTEGRABILITY
505
Let Z\, Z2, ■.. be a sequence of i.i.d. U-valued random variables with common
law m living on a still different probability space (Q2> -^2. Pi)- Exercise 1.24 and
Fubini's theorem imply that (11.3.3) is equivalent to
supsuprr1/pr1/Q|/(TniZj)| < oo P, x P2-a.s. (11.3.4)
n>lj>l
This is the crucial relation. To derive, say (11.3.2), use (11.3.4) and Fubini's
theorem to conclude that for P2-almost every choice of Zi, Z2, ■ ■ ■,
supn-'/"(supj-|/o|/(T„,Zj)|) < oo Pra.s..
n>l . j>l
Therefore, by Exercise 1.24, for every such Z\,Z2,...,
co > £i(sup(T,/Q|/(Tn,ZJ-)|))P
Jtj>ix '
> supj-^ f \f{ttZ^u{dt).
j>\ JT
Applying Exercise 1.24 once again, we obtain
/ (J \f(t,x)\*v(dt)y/Pm(dx) = £2(£|/(t,Z,)|M*))
proving (11.3.2). The proof of (11.3.1) is identical. I
Remark. It turns out that both expressions in (11.3.1) and (11.3.2) play an
important role in the distribution of the integral JT \X(t)\pv(dt) when it is finite.
We will return to this point in the sequel.
Let log+ x = log x if x > 1 and 0 otherwise.
Theorem 11.3.2 Let {X(t), teT} be a measurable a-stable process, 0 < a <
2, with an integral representation (10.1.1). Ifa= I, we assume that the random
measure M is symmetric. Let p > 0. Then JT \X{t)\pv(dt) < oo a.s. if and only
if
J \r)(t)\pv{dt) < oo (11.3.5)
and one of the following conditions holds:
a/p
< CO,
f ( f \f{t,x)\am{dx)\ v(dt) < co when 0 < p < a; (11.3.6)

506 MEASURABILITY, INTEORABILITY AND ABSOLUTE CONTINUITY 11.3
I / \f(t,x)\a [\.+ log+A(t,x)]m(dx)u(dt) <oowhenp = a, (11.3.7)
where
= \f(t,x)\"fTfE\f(u,vTm(dv)Hdu)
^ ] jE\f{t,v)\°m{dv)jT\f{u,x)\°v{duy
j ( / \f[t,x)\pv{dt)\ m(dx) < oo when p > a. (11.3.8)
PROOF: We will not give a proof in the case p = a because the one we know
makes use of facts that fall outside the scope of this book. We refer the reader to
Rosinski and Woyczyriski (1986) and Kwapieri and Woyczyriski (1987) instead.
Suppose p ?£ a, and assume firstly that 77(f) = 0, and M is a SaS random
measure.
Case 1. 0 < p < a. The necessity of (11.3.6) follows from Proposition 11.3.1.
On the other hand, (11.3.6) implies
E j \X(t)\*v{dt) = Ca,pJ (J \f(t,x)\am(dx)Y u(dt) < oo,
where Ca# is a positive constant depending only on a and p. Thus
fT \X{t)\pu(dt) < 00 a.s.
Case 2. p > a. The necessity of (11.3.8) follows once again from
Proposition 11.3.1. On the other hand, suppose that (11.3.8) holds. Then f(-,x) e
LP(T, T, v) for almost every x € E and (assuming once again that m is a
probability measure),
£||/(-,Z)ll!»(r.7>)<«>l (11.3.9)
where Z is an jB-valued random variable with law m. Let Z\, Z2, ■ ■ ■ be i.i.d.
copies of Z. Consider
00
5(f) = ^€ir-1/a/(t,^),feT,
where {tj, j > 1} and {Tj,j > 1} are independent sequences of, respectively,
Rademacher random variables and arrival times of a unit rate Poisson process,
independent of the sequence {Zj, j > 1}. We know that the series S(t) converges
a.s. for every t € T, and that {5(f), f £ T} is a measurable version of {X(t), t €
T} up to a multiplicative constant.
It is now sufficient to show that the series
00

11.3
CONDITIONS FOR INTEGRABILITY
507
is a well-defined element of LP^T, T, v), i.e., it converges a.s. in Lp(T,T,u).
Indeed, S e Lp(T,T,v) a.s. means JT \S(t)\pv(dt) < oo a.s. and, since
{S{t), t e T} is, up to a constant, a measurable version of {X(t), t € T},
we have JT \X(t)\pu(dt) < oo a.s. as well by Theorem 11.2.1. Consider two
cases:
1. Case a < p < 1. Then
JT j=N JTj-N
CO
= E r7P/aU/(-> ^)IIlp(t.t,w - ° a-s- ^ iV - oo,
by (11.3.9) and Theorem 1.4.5.
2. Case p > 1. We can assume that the sequences {rj, j > 1} and {.Z,-, j > 1}
are defined on a probability space (Qi, T\, P\) and the sequence {gj, j > 1} is
defined on a different probability space i^li^^Pi)- By Fubini's theorem it is
enough to prove that for P\ -almost any u>\ S Qi the series of independent random
variables
oo
^rj^o/o.^u,,))
converges P2-a.s. in LP(T, T, v). By the Ito-Nisio theorem (It6 and Nisio (1968)),
it is enough to prove that we have L1 convergence in LP(T, T, v), i.e.,
M
fcco^(|5:eiI7,/>>)/(-.^(««'0)|LF(riTJ =0 P,-a.,
j=N
(11.3.10)
Suppose, firstly, 1 < p < 2. Then
M
\;T-l/a(u,)f(:ZAu>,))\\
»(T,7»
j=N
< (E2 [ \j2eJrjl/a(u>)f(t,Zj(uJl))\'v(dt))1/P
"'T j=N
(Jensen's inequality, p > 1)

508 MEASURABILITY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.3
M
(Jensen's inequality, p < 2)
= (/T(E Tfa{u>,)f{t,Z^x)f)P'\{dt))X
J2rJP/aMf&ZM))\Pv(dt)) " (sincep<2)
M (
= (EI7P/a(w«)»/(-.zj(wi))IIW.r.,))
ri-'-(o;0H;(.Izj^1))||2P(TiTil/)j1/P
which tends to zero Pra.s. as M, TV -> oo, by (11.3.9) and Theorem 1.4.5.
Ifp>2,
M
Hl^r'"Mn-,^,))\i(TTj}
j=N
. M II II
<Up) (j2Tfa^f^zM))2) "(dt)) p
(Khinchine inequality: Exercise 11.1)
< c{vf'p £ (/ r-p/>,)|/(t, zMW»m)
(Minkowski's inequality for p/2)
M
- <p)Vp £ rj2/aM||/(-, zMMUt,^
which tends to zero Pi-a.s. as M, N —> oo, again by (11.3.9) and Theorem 1.4.5.
This proves that the series S converges a.s. in L?(T, T, v), and thus the proof of
the theorem is complete in the symmetric case.
In the general case, but still assuming r)(t) = 0, let {X\(t), t £ T} and
{Xi{t), t e T} be two independent measurable copies of {X(t), t € T}. Then
{Y{t) = 2-1/Q(Xi(t) - X2(t)), t e T} is 5qS with an integral representation
(10.1.1), but this time the random measure M is symmetric and has the same
control measure m as before. Now our claim follows from the easily verifiable
fact that JT \X{t)\Pv(dt) < oo a.s. if and only if JT |y(t)|pi/(cft) < oo a.s. (see

11.3
CONDITIONS FOR INTEGRABILITY
509
Exercise 11.2). Finally, the sufficiency part of the theorem in the case ;/(') = 0
implies the necessity of (11.3.5). I
Remarks
1. The preceding proof shows actually that JT \X(t)\pi/(dt) < oo a.s. if and
only if (11.3.4) and (11.3.5) hold. The role of (11.3.4) is especially
interesting. It is equivalent to /gsupn>| n~^v\f{Tn,x)\am(dx) < °°
(Relation (11.3.3)), for Pi-almost every choice of tut2,... (using the
notation of the proof of Proposition 11.3.1). If a > 1, this does not
imply P{limn_00n-|/pX(rn) — 0} = 1 for any choice of T\,Ti,...
for which (11.3.3) holds. Of course, JT \X{t)\pv(dt) < oo implies
P(limn^00n-^pX(Tn) = 0) = 1 (Exercise 1.24) only for a P,-a.s. choice
°f Ti! r2, • • • • Although (11.3.3) is true for a Pi-a.s. choice of T\, T2, ■ • •
as well, there may be realizations tj,t2 ... for which (11.3.3) holds but
P{\imn^00n~l/pX{Tn) = 0) = 1 does not hold. This seems,
nevertheless, to suggest that, even when a > 1, there are "not too many cases" in
which the converse to Theorem 10.3.1 fails.
2. Although proven in the real-valued case, Theorem 11.3.2 remains obviously
true for measurable complex-valued SaS processes.
3. In the case a — p, one can express (11.3.7) as
|/(i, a:)|Q [l + log+ B{t, x)] m{dx)v{dt) < 00, (11-3.11)
JT JB
IT JB
where
B(t,x)
fE\f{t~v)\am(dv)JT\f{u,x)\°v(dv.y
because for any constant C, there is a constant a depending only on C such
that I log+ Cu — log+ u\ < a for any ti£l.
Example 11.3.3 Shepp (1966) proved that if {X{t), t > 0} is Brownian motion,
and if q(t), t > 1 is a measurable function such that infi<t<T q(t) > 0 for all
r > 1, then, for any 0 < p < 00,
/•-TOP
y, ?w<
if and only if
r°° iv/T-
I W)dt<0°-

510 MEASURABILITY. INTEGRABILITY AND ABSOLUTE CONTINUITY 11.3
We can now obtain a corresponding result in the a-stable case. Let {X(t), t > 0}
be an a-stable L6vy motion, 0 < a < 2, X(t) = /„' M{dx) where M has
Lebesgue control measure. Assume also that M is SaS when a = 1. Let q(t)
be a non-negative function such that the set {t: q(t) = 0} has zero Lebesgue
measure. The measure u with density v{dt) = q(t)~xdt is thus cr-finite. Then,
for any T C [0,oo),
|X(i)|*
dt < oo a.s.
if and only if
a/p
dx < oo when p > a
and,
It W)dt < °°'
/tL/o log+(*Jrn(*,oo) ^)-1^]^) < ~.
when p = a.
Thus, taking g(t) = f, we get for any 0 < a < 2 and p > 0,
Jf
' ^ '' dt < oo a.s. if and only if 77 > - + 1
and
/
Jo
' ^' -dt < 00 a.s. if and only if 77 < - + 1.
,0 t" a
Example 11.3.4 Let {X(t), t 6 R} be a measurable version of the well-balanced
SaS linear fractional stable motion of Example 3.6.5. Let
Y(t) = f e~u{X(t) - X(t + u))du, t g R.
Jo
It is clear that {Y (t), t € R} is a stationary SaS process w/ie/iever the integral is
denned. Theorem 11.3.2 shows that the latter is true for any 0 < H < 1 if a > 1,
but only for £ - 1 < H < 1 if j < a < 1, and for no H if a < 5.
Example 11.3.5 Ltt{X(t), t 6 R} be a measurable version of the harmonizable
SaS process (6.5.1). It follows from Theorem 11.3.2 and the above remark that
for any T > 0, and -00 < a < b, the integral
/.
T „—ita _ p—itb
-^—X{t)dt
it

11.4
CHANGING THE ORDER OF INTEGRATION
511
is well defined. This suggests an a.s. inversion formula for the Fourier transform
of a SaS noise. We will return to this point in the sequel.
11.4 Changing the order of integration
Let {X(t), t € T) be a measurable a-stable process 0 .< a < 2 such that
JT \X(t)\v(dt) < oo a.s. What is the integral $TX(t)v(dt) equal to? Let p.
be a probability measure equivalent to v, h = dv/dp, and let t\,T2,... be a
sequence of i.i.d. T-valued random variables with common law p, defined on
a probability space (Q], T\, Pi), different from the probability space (O, T, P)
where the process {X{t), t € T} is defined. By the strong law of large numbers,
for P-almost every w6fl,
Pl (li,^1^1^;-^^1-^ jU*M*)) = i.
and thus, by Fubini's theorem, there are (non-random) points t\, t2,.. .inT such
that
Pf lim x(uMtl) + ... + x{tn)k(tn) = r a = L
Since n~l(X(ti)h(ti) + • ■ • + X(tn)/i(t„)) is an a-stable random variable for
every n = 1,2,... (Theorem 2.1.2), the integral jTX(t)u(dt) is an a-stable
random variable as well. What are its parameters? They are easy to find if
{X(t), t S T} has an integral representation (10.1.1) andifwe may interchange
the deterministic and stochastic integrations. The following theorem, which is a
Fubini-type result, justifies such a change of order of integration in the case where
T is a Borel cr-algebra on a separable metric space (T, p).
Theorem 11.4.1 Let {X(t), t € T} be a measurable a-stable process on a
separable metric space (T, p), given by
X{t) = f f{t,x)M{dx) + r){t), t 6 T,
Je
where M is an a-stable random measure with control measure m. Assume that
M is symmetric when a = 1. If fT \X(t)\i>{dt) < oo a.s., then
f X{t)v(dt) = f ( f f{t, x)v(dtU M{dx) + J rj(t)u(dt) a.s.
Thus, in particular, JT f(t, -)v(dt) £ La{E, £, m).

512 MEASURABIUTY, INTTEGRABILITY AND ABSOLUTE CONTINUITY 11.4
Remark. We are assuming that {X(t), t £ T} is defined a.s. by its integral
representation because the conclusion of Theorem 11.4.1 is an a.s. result. Of
course, if (10.1.1) is only a distributional representation of {X(t), t e T}, then
the conclusion of Theorem 11.4.1 should also be understood in the sense of
equality of distributions.
PROOF OF THEOREM 11.4.1. We may and will suppose r](t) = 0. Suppose
firstly that a > 1. Assuming, as usual, and without loss of generality that v is
a probability measure, we choose a sequence of i.i.d. T-valued random variables
t\ , T2,... with common law u, defined on a probability space (Q.\, F\, Pi) different
from the probability space (Q, T, P) on which the process {X(t), t £ T} is
defined. By the strong law of large numbers, for almost every w£Q,
lim *(T.) + -+*(Tn)as ; Xmdt)pr^
i—»oo n
= / X{t)u(dt)
Jt
Therefore, by Fubini's theorem, for Pi-almost every lo\ € Q\,
I
X{t)v{dt) = lim -^TxItAuA) P-a.s. (11.4.1)
n—*oo 71 ^—J
i-\
Let us now consider the integral fE(fT f{t, x)u{dt))M(dx). We may regard
{/(tj, x) x € E}, j — 1,2,..., as a sequence of i.i.d. random vectors taking
values in a separable closed subspace S of the Banach space La(E, £, m). (See
Theorem 11.1.1.) The norm of these random vectors has a finite mean because
25,11/(7*,*), x G £||LQ(E,£,m) = E,[j \f[rhx)\am{dx))"a
= I ([ \f{t,x)\am{dx)\ av(dt) < oo
by Proposition 11.3.1. We can therefore apply the strong law of large numbers for
5-valued random vectors to obtain
{ j f{t,x)v{dt), ie£} = Ex{J{tj,x), xeE}
1 °°
= lim - V {/(t,,x), x e E\ Pi-a.s.
n—>oo n *■—'
in 5 and thus in La (25, £, m). In other words, for Pj -almost every u)\ e &i,
lim-f]/(Tj(a;i),-)= / f{t,-)v{dt) inLa(E,£,m),

11.4
CHANGING THE ORDER OF INTEGRATION
513
and thus, by Proposition 3.5.1, we obtain
J (| f(t,x)v{dtj)M{dx) = ^J {^J2f(rM)^))M(dx)
= }^i^x{Tj{Wi))- (11A2)
Since both (11.4.1) and (11.4.2) hold on the u>i-sets of measure 1, there is an
W[ € Qi for which both of them hold. This completes the proof in the case a > 1.
Consider now the case 0 < a < 1. Since the control measure m is essentially,
c-finite (Theorem 11.1.1), we may assume that m is actually a probability measure.
We will use the following "randomization" lemma due to Kallenberg (1988).
Lemma 11.4.2 Let S and S' be, respectively, a separable metric space and a
Polish (i.e., a complete separable metric) space, and let f:S'^>Sbea Borel
measurable function. Let C and 6 be, respectively, an S- and a S'-valued random
variables defined on a probability space (Q., jF,P).]f£ — f{6), then there is an
S1-valued random variable 9' on an extended probability space (Q x [0,1], T x
B, P x Leb) such that 6' = 6andC,= J{9') Px Leb a.s.
This lemma shows that the equality in distribution £ = f(8) can be realized
as an a.s. equality. To complete the proof of Theorem 11.4.1, we apply Lemma
11.4.2 as follows.
Let 5 = Ll{T,T,u) x E and S' = R°° x E» x {Ll(T,T,u))°° x K°°.
Observe that under our assumption the space LX(T, T, v) is separable (Dun-
ford & Schwartz 1958, Vol 1, Lemma I1I.8.3).1 Therefore, both S and S" are
Polish spaces when endowed with product topologies. Let £ = (Ci^). where
d = {X(t), i 6 T} and (2 = jT£(/T/(t,x)i/(dt))M(dx). Note that C2 is
a well-defined (real-valued) random variable because Theorem 11.3.2 implies
Ie I It f{t,x)v{dt)\am(dx) < 00. Hence, C is a well-defined 5-valued random
variable.
LetnowTi,!^,.. .be a sequence of arrival times of a Poisson process with unit
rate independent of the sequence of i.i.d. random vectors (^), (^),..., where
VI, Vi, • • • are E-valued with common law m, and
P(7j-l|Vi) = 1-P(7J = -W) = i±fM-
1 Because T is separable, we can choose a countable dense set To and approximate a function /(x)
inL1(T,T,i') by simple functions of the form £"=1 ail(||X - Ci|| < r<)> "^ 1, where the Cis
are in To and the a^s and TjS are rational.

514 MEASURABILITY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.4
Here, B is the skewness intensity of the random measure M. Define an 5'-valued
random variable 8 = (8U 82,83,84) by
#1 = (71.72, •••).
82 = (r,,r2,...),
03 = ({f(t,Vj),teT},j=l,2,..-),
04 = {J f(t,Vj)v(dt), j = 1,2,...).
Let /: S" -> S be defined by /(xt, x2,x3)X4) = (y,, y2), where
CXJa £°1, x\ U)xiU)~l/a^U) if *e sum converges in Lx (T, T, u),
yi
otherwise,
CxJa £°1, xi {J)x2(.j)~x,aX4(j) if the sum converges,
2/2= ;
otherwise,
and Ca is given by (1.2.9). Observe that / is clearly a measurable function
and, moreover, in the proof of Theorem 11.3.2, we established that the series
E^=i7jr71/Q/(-.^) converges a.s. in L\T,T,v). Corollary 3.10.2 and
Exercise 11.3 therefore imply < = f(8).
Applying Lemma 11.4.2, we conclude that there are two independent random
sequences, r\, T2, ■ ■ ■, and
such that (r',, T2,...) = (Ti, r2)...); I sx | ( s2 | ,... are i.i.d. random
V 7i
vectors, the Tj are Lx (T, T, u)-valued and
. /{/(^i),«€T}
= JTf(t,VMdt) |. (H.4.3)
% I \ Ij
These sequences are such that
{X(t), teT} = {C^f^-^r'^t) t 6 T} z.s.m L\t,T,v)
(11.4.4)

11.5 TAIL BEHAVIOR OF THE Lp-NORM DISTRIBUTION 515
and
Je (jT/(t,x)i/(d*)) M{dx) = C^f^^r^sj a.s. (n.4.5)
Relation (11.4.4) yields
jf X(*M<ft) = C^a y J2ii{T'j)-Varj(t)u{dt) a.s. (11.4.6)
Note that
00
/ Ea^-'^WK*) = / Er71/ai/(f^i)i^(di)
1/7 j=i •/rj=i
= Er7'7 / l/(*,Vri)K<fc)<ooa.s.
by Theorems 11.3.2 and 1.4.5. Applying Fubini's theorem, we obtain by (11.4.3):
[ x(t)»(dt) = c^^mr^ [ rj(t)u(dt)
Jt j=1 Jt
00
= ^^7}(n)-'/aSja.s.
This, together with (11.4.5), concludes the proof of the theorem. 1
11.5 Tail behavior of the I^-norm distribution
We consider now the asymptotic behavior of the distribution of the "ZAnorm"
i/p
Mxy* = (JT\x{t)\*u{dt)
Consider two cases, p > 1 and 0 < p < 1. When p > 1, {JP{X))^P is a norm
on the sample paths of {X{t), t 6 T}. Corollary 6.20 of Araujo and Gine (1980)
implies that, if JP(X) is finite,
lim^X"P(Jp(X)x/p >X) = Caf (J \f{t,x)\pu{dt)\ m{dx),
(11.5.1)
where Ca is given in (1.2.9). Compare this limit with condition (11.3.2) in
Proposition 11.3.1.
The next theorem states that (11.5.1) remains true even when 0 < p < 1.

516 MEASURABILITY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.5
Theorem 11.5.1 Let {X(t), t GT} be a measurable a-stableprocess, 0 < a <
2, with an integral representation (10.1.1). Ifp > 0,thenJp(X) < oo a.s. implies
(11.5.1).
PROOF: This theorem follows from Rosinski and Samorodnitsky (1993). Here,
we prove it under some slightly stronger assumptions. We shall assume that the
control measure m is finite and there is an e > 0 such that JP(X') < oo a.s.,
where
X'(t) = / f(t,x)M'(dx) + V(t), t € T,
JB
and ft € La+€(E, S, m). Here, M' is an (a + e)-stable measure on (E, £) with
the same control measure and skewness intensity as M.
Note that JT \X'(t)\pv(dt) < oo a.s. implies JT |X(t)|pi/(di) < oo a.s. and
hence JP{X) < oo a.s. This is because (11.3.4) holds with a if it holds with a + e.
(The conditions (11.3.4) and (11.3.5) are necessary and sufficient for integrability,
as noted in the remark following Theorem 11.3.2.) Of course, we may and will
assume that m and u are probability measures. We may also assume 7?(i) = 0.
By Corollary 3.10.2,
|5>ir;1/a/(t,vj) -aj(a,t))\\(dt)) \
- i=i
(11.5.2)
using the notation of Section 11.1. The function a,-(a, t), in particular, is defined
in (11.1.2). Consider separately
wl = cUa[J |7irr,/a/(t,vl)-o,(Q,t)|Mdt))l/P (n-5-3)
and
W2 = ClJa(J |f>ir-,A7(^-) - aj(a,t))\%(dt))l/P. (11.5.4)
It follows from Theorem 11.3.2 that W\ < oo a.s„ and thus W2 < oo a.s. as well.
We want to show that it is W\ that determines the asymptotic tail behavior of
JP(X). We have, by Exercise 1.22,
lim XaP{Wx > A)
A—>oo
= lim^x" p(c!Ja J |7irr1/a/(t,v'1)|Pt/(di))1/P > a)
= ca ^Apfr, < a-1 [J |/(t, v,)|Md*)]a P)

11.5
TAIL BEHAVIOR OF THE Lp-NORM DISTRIBUTION
517
= CaE(J' \f{t,Vi)\"Hdt)y/P
= C"J(J \f(t,x)fv(dt)y/Pm(dx).
If we prove
EW? < oo,
(11.5.5)
(11.5.6)
then our theorem will follow from (11.5.2), (11.5.5) and (11.5.6).
Let {if >, rf,..., V,W, V2(i),..., 7fj), 7«,...}, i = 1,2, be two
independent copies of the random variables determining W2 and let
W? = <#"( / IXW-,/a/(t,lf) - aJ(a,t)|PKA)),/'1 i - 1,2.
It is clearly enough to prove
and
£|W2(1)-W2(2)|Q <oo if p> 1
E\{W^Y - (W2(2))pjQ/p < 00 if 0 < p < 1.
(11.5.7)
We shall treat the case p > 1. (The case 0 < p < 1 is identical.)
Let ei, €2,... be a sequence of i.i.d. random signs independent of all the other
random variables. Choose a positive integer m so large that a/pm < 1 and let
AJ,, = 7?)rS,J-^/(*,v;,>)-7f)rf)-,/a/(t,^2)).'
Then
E\W^-wW\a
(00 \ a/p
/ i5Zi4^*ri'^ ] ^y Minkowski-p ^^
JT\iLeJAj4u(dt))
< CaEltT,v
= CaE1jy
Et
M'
-1 a/p-n
ejAjj
v{dt)
(Holder, a/pm < 1)
a/pm
f & iiE £j^.'* Pj/(*') ■" ■ v^dtm)
JT,x-xTm fc=1'j_2

5! 8 MEASURABILITY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.5
< CaEyJ,v
TO OO \
Y[Et\YJ^AjMVm\v{dU)---u{dt
- a/pm
(by Holder)
< CE
CE
< CXE
I
m oo
TiX-xTm
a/pm
Il(E4tk)P "(dti)-""(dU
(by Exercise 11.1)
/r(E4.)"!^')
i=2
jT{fyfas^viY)p,\{dt) P„
where C and C\ are positive constants.
Now, let {X'^t), t G T}, i — 1,2, be independent measurable copies of
{JT(i), t € T}. Then {Y(t) = 2-'(°+0(x((t) - X^(t)), i S T}, is a
measurable S(a + e)S process with an integral representation (10.1.1) where the random
measure M has the same control measure m as before, but M is S(a + e)S.
Clearly, JT \Y{t)\Pu{dt) < oo a.s. By (11.3.3),
L
suV(n-Vp\f{Tn,x)\)a+em{dx) < oo
E n>l
(11.5.8)
for almost every choice of i.i.d. T-valued random variables t\,tz,.. . with common
law v. Fix ti,t2, ... for which (11.5.8) holds. Then Eg(V\)a+€ < oo, where
g(x) = supn>1 n~1/p|/(Tn,x)|, x € E, and Vi is as above. Therefore, letting
6[, 62, ■ ■ ■ and Ti, T2,. • ■ be, as above, independent sequences of i.i.d. random signs
and Poisson arrivals, respectively, independent of the i.i.d. sequence Vi, Vj,...',
we conclude that
oo
B|£^r71AW-)| <oo.
5=2
Applying Khinchine's inequality yields
oo>E|5>r7I/as(yi)| > const. EfjTr-^giVj)2)
j=2 j=2
oo
> const. E(supn-VpYlTi2/af(Tn,Vj)2)
a/2
,°"/2
Vi>l
j=2

11.6
INVERSION FORMULA FOR HARMONIZABLE SaS PROCESSES
519
We conclude by Exercise 1.24 that
oo
supr2/° supn-2>p J2(Tf)-2/af(rn, vf >)2 < oo a.s.
i>i n>\ j=2
where {Tf ,V^\ j = 1,2,...}, i = 1,2,..., are i.i.d. copies of {n,,Vj-, j =
1,2,...}, independent of the sequence n, t2, .... By Fubini's theorem, for almost
every choice of {if \ yf \ j = 1,2, ...},*= 1,2,...,
OO
supn-2/"(supi-2/Q ^rii)_2/Q^(T"' VPf) < °° a-S"
n>l M>1 ~
- - ]=2
and thus, by Exercise 1.24,
oo
oo > EMupi-^Y^^fi^V^)2)
J=2
v7>/2
> sup,-"/° / (JTrf-2/q/(*, V*')2) "(<*')•
l— ^ 1=2
Applying Exercise 1.24 again, we conclude that
< oo,
proving (11.5.7). This completes the proof of the theorem. I
11.6 Inversion formula for harmonizable SaS
processes
We shall use the results of the previous section to derive an inversion formula for
harmonizable SaS processes, 0 < a < 2.
Let M be a complex-valued SaS random measure on (R, B) with a finite
circular control measure k and control measure m. Let
r°°
Z(t) = / eltxM(dx), -co < t
J —OO
< OC.
and
/oo
eitxM(dx), -oo < * < oo,
'OO
be the complex and real SaS harmonizable processes defined, respectively, in
(6.5.1) and (6.2.13). It is appealing to regard these wo processes as Fourier

520 MEASURABILITY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.6
transforms of the random measure M. If this view is justified, then we would
expect these processes to satisfy a kind of inversion formula similar to the one for
Fourier transforms of non-random measures. The next result shows that such an
inversion formula does exist.
Theorem 11.6.1 Let {Z{t), -co < t < oo} and {X(t), -co < t < co} be the
complex and the real SaS harmonizable processes. Then, for any —oo<u<
v < oo,
11 i rT P—iut — p—ivt
-M({u}) + -M(W) + M((u,t;)) = Plim^^- ^ e- -^ Z(t)dt
(11.6.1)
and
^M({-u}) + iM({-t;})+M((-t;,-U)) + iM({«}) + iM(M)+M((«,w))
1 fT e~iut — e~ivt
= plimT^00-/ X(t)dt. (11.6.2)
7T J_x it
Remarks
1. It is implicitly assumed that (11.6.1) and (11.6.2) involve measurable
versions of the processes {Z(t), -oo < t < oo} and {X(t), -oo < t < oo},
respectively.
2. Example 11.3.5 implies that the integrals in the right-hand sides of (11.6.1)
and (11.6.2) are well defined for every T > 0.
PROOF OF THE THEOREM: Theorem 11.4.1 (in fact, its counterpart in the complex-
valued case) implies
i rT —iut _ „—ivt p+oo
T" / ~ T- Z(t)dt= fT(x)M(dx), (11.6.3)
27T J_T it J_00
where
1 rT p»(z—«)* _ „i(x-v)t
Wl-si, 5 * (11'6-4»
It is well known (and easy to verify) that there is a finite constant K such that
|/r(aOI < K f°r every T > 0 and a; £ R, and, moreover,
f(x) = lim fT(x) = -l{u}(z) + z!{»}(«) + l(u,»)(*)-
X—*oo Z Z
Proposition 6.2.3 and the bounded convergence theorem imply
i rT p-iut _ p-ivt poo
P»mr-.oo^ J_ ~ -rf- Z{t)dt = J f(x)M(dx),

11.6 INVERSION FORMULA FOR HARMONIZABLE SaS PROCESSES 521
proving (11.6.1).
For the second part of the proof, we again apply Theorem 11.4.1 and obtain
u
T p—iut p—ivt
A—x{t)dt
where
(i), . 2 fT sinvt — sinut
g>r'(x) = — / - cos xtdt,
/oo /»oo
g^\x)M{dx)-ilm g$\x)M(dx), (11.6.5)
-oo J—oo
>(*)=\ f
7T Jo
(2) , . 2 f COS Vt — COS Ut .
gq, {x) — — / sinxi dt.
n Jo t
See Exercise 11.4. It follows from the corresponding properties of the functions
fr in (11.6.4) that g)j? and g? are uniformly bounded (over x and T),
ff(')(x) = lim (#>(!)
T—»oo
= ^wt1) + 1(-».-")(a;) + 21<-"}^ + ^M^)
and
^(s) = lim g$\x)
1 —+00
= 2l{~v)^ + 1(-v--")^ + 21{-"}^ ~ 21{n}^
~ l(U)„)(a:)--l{„}(a;).
See Exercise 11.5. Proposition 6.2.3 and the bounded convergence theorem imply
1 fT e~iut — e~ivt
plimT_-y_7 X(t)dt
/oo /*oo
gW(x)M{dx)-ilm g(-2\x)M{dx).
-oo J —oo
This proves (11.6.2) and concludes the proof of Theorem 11.6.1 1
Corollary 11.6.2 Let M\ and M% be complex-valued SaS random measures on
(71, B) with finite circular control measures k\ and ki- Let
/oo
eitxMj(dx), -oo<t< oo,
■OO

522 MEASURABILITY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.6
and
/oo
eitxMj(dx), -co < t < oo,
•oo
j = 1,2, be the corresponding complex and real harmonizable processes. Then
{Zi(t), -oo < t < oo} = {Z2(t), -oo < t < oo}
if and only ifk\ = k2. Moreover,
{Xi{t), -oo < t < oo} = {X2(i), -oo < i < oo}
i/and on/y j/fci + fci = fc2 + &2 on E\{0} (w/iere fy, j - 1,2, w de/med by
(6.5.2)) and /ft |«,|a*,({0} x <fe) = /ft |Sl|QM{0} x ds).
Proof: Clearly, fc, = fc2 implies {Z,(t), -oo < t < oo} = {Z2{t), -oo <
£ < oo}. For the converse, note that Theorem 11.6.1 implies that for any -oo <
u < v < oo,
M, ({«}) + M, ({«}) + 2M, ((u, v)) = M2({u}) + M2(M) + 2M2((u,«)).
If it and v are continuity points of both control measures rrij(-) = fcj(- x S2), j =
1,2, then Mi ((u,u)) = M2((u,v)), and hence
M(u,v) x A) = fc2((u,u) x A) (11.6.6)
for every Borel set A C S2. Since such continuity points u and v are dense in E,
we conclude that (11.6.6) holds for every —oo < u < v < oo and Borel A Q S2,
and thus fci = k2.
To prove the second part, suppose firstly k\ + k\ = k2 + k2 on all of M. Let
Mj be a complex-valued SaS random measure on (R, B) with circular control
measure k j, independent of the random measure Mj, j = 1,2, and define
/oo
e"xMj(dx), -oo < i < oo, j = 1,2.
-oo
It is easy to verify that {Yj(t), -oo < t < oo} = {X,(t), -oo < t < oo},
j = 1,2. Therefore, for every j = 1,2,
{X,(t), -oo < t < oo} = {2-xla(Xj(t) + y,(f)), -oo < t < oo}. (11.6.7)
But, for j = 1,2,
{X,(t) + Yj{t), -oo < t < oo} = {Re /" eitxMj(dx), -oo<t< oo},
(11.6.8)

11.6 INVERSION FORMULA FOR HARMONIZABLE SaS PROCESSES 523
where Mj is a complex-valued SaS random measure with circular control
measure kj + kj, j — 1,2. Relations (11.6.7) and (11.6.8) imply {X{(t), -oo < t <
oo} = {X2(t), —oo < t < oo}, because k\ + k\ = k2 + k2.
If we only have kx + fc, = k2 + k2 on M\{0} and J^ |si|QJS;i({0} x ds) =
/s \s[ \ak2({0} x ds), we obtain instead
JRe j eitx 1 {x ^ 0}M, (dx), -oo < t < oo]
eitxl{x ^ 0}M2(dx),-oo < t < oo] (11.6.9)
^ J —oo
and
ReM,({0})=ReM2({0}). (11.6.10)
But the left-hand (right-hand) side of (11.6.9) is independent of the left-hand
(right-hand) side of (11.6.10) and the two sum to {-X'i(t), -co < t < oo}
and {X2(t), -oo < t < oo}, respectively, so (11.6.9) and (11.6.10) yield also
{Xi{t), -oo < t < oo} = {X2(t), -co < t < oo}.
To establish the converse statement, note that Theorem 11.6.1 implies that for
any —oo < u < v < oo,
M,({-u}) + Af,({-«}) + 2F,((-«, -u)) + Af,({«}) + M,(H)
+ 2M, ((«,«))
= M2({-u}) + M2({-v}) + 2M2((-v, -u)) + M2({u}) + M2({v})
+ 2M2((u,v)). (11.6.11)
If 0 < u < v are such that the four points u, —u, v, —v are continuity points of
both rn\ and m2, then
M,((-v, -«)) + M,((u,t;)) = M2((-w, -«)) + M2((«,t;)),
so for zx, t; as above,
(fc, +h){(u,v) xA) = (k2 + k2)((u,v) x A) (11.6.12)
for every Borel set A C S2. Since points u, v as above are dense in (0, oo),
(11.6.12) holds for every 0 < it < v and every Borel set A C S2. Since
(kj +kj)((-v,-u) xA) = (kj+ kj)((u,v) x A*),
(see (6.5.2)), (11.6.12) extends also to—oo < u < v < 0. Thus, fci+fci = k2 + k2
on R\{0}. Finally, choosing 0 = u < v in (11.6.11), we obtain Re Mi({0}) =
Re M2({0}), implying /ft |5,|a*,({0} x ds) = JSj |s,|Qfc2({0} x ds). ■

524 MEASURABILITY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.7
11.7 Absolute continuity of stable processes
We want to find conditions for a stable process {X(t), t G [a, b]} with integral
representation (10.1.1) so that there is a version {Y(t), t G [a, b]} whose paths
are absolutely continuous. The derivative
will then exist at almost every t G [a, b]. We also want to ensure that for a given
V > 1, Ja l^(*. w)lPdi < °° for a11 w-
Let us start with a formal definition of absolute continuity.
Definition 11.7.1 Let -oo < a < b < oo be two real numbers. A function
4>: [a,b] —► K is called absolutely continuous (we write cj> G ^4C([a,b])) if for
every e > 0 there is a 6 > 0 such that for any family (uk, vk), k = 1,2,..., of
disjoint intervals in [a, b],
m m
]T|iifc-ufc| <<5 =>> 52|0(vfc) — ^(u*:)| <e.
fc=i fc=i
It is a well-known fact that 0 is absolutely continuous if and only if there is a
function <fi e Lx [a, b] such that
4>{t) = <t)(a)+ j <p{s)ds, te[a,b]. (11.7.1)
./a
An absolutely continuous function is, therefore, continuous on [a, b] and differen-
tiable at almost every point t £ [a, b). More precisely,
~dT ~ 0(t)
for almost every t G [a, b}.
Definition 11.7.2 ACp[a, b] is the set of all absolutely continuous functions <j> on
[a, b] such that ^> € L^a, 6], p > 1.
Observe that j4CP[a, 6] is a linear space. The following proposition is valid
for general stochastic processes, not necessarily a-stable.
Proposition 11.7.3 Let {X(t), a < t < b} be a stochastic process that has a
version with all sample paths in ACp[a, b], p > 1. Let {Y(t), a < t < b}
be a version of {X(t), a < t < 6} with continuous sample paths, defined on a
probability space (Q, T, P). Then there is an event Qq € T with P(Ch) = 0

11.7
ABSOLUTE CONTINUITY OF STABLE PROCESSES
525
such that for every u> € Q.q, the function Y{-,u): [a, b] —> R is in ACp[a, b] and,
moreover,
Y(t,cj)=Y(a,oj)+ j D(s,oj)ds, a<t<b, (11.7.2)
Ja
for some measurable stochastic process {D(t), a <t <b} with all sample paths
in Lp[a, b]. Moreover,
D(t,uj) = —Y(t,ui) (Leb x P) - a.s.
at
PROOF: Let {Yi(i), a < t < b} be a version of {X{t), a < t < b} with
all sample paths in ACp[a,b], p > 1, defined on a probability space (Q,T, P).
Then, every sample path of {Yi(i), a < £ < 6} is continuous and, for j = 1 and
every e > 0,
m
lim Pf sup supy^\Yj{vi) - YAm)] > <■) = 0 (11.7.3)
— i=i
where the second supremum is over all m, vi rational, i = 1,..., m, such that
m
a < u\ < v\ < v.2 < V2 < ■ • • < um < vm < b, and Y^ \vi - Uj| < 6.
i=l
To determine whether the converse holds, let {Y2(t), a < t < b} be any other
version of {X(t), a <t < 6} with all sample paths continuous (and defined, for
simplicity, on the same probability space (Q, T, P)). Then (11.7.3) holds with
j — 2, and the continuity of the sample paths of {Y^Wj a < t < b} implies
that, apart possibly from a null set, the realizations of {Yi(t), a < t < b} are
in AC[a,b\. We want to show that, apart from a null set, the realization of
{Y2(t), a < t < b} are, in fact, in ACp[a, b] as well. Define for j = 1,2,
A< = I(t,w) e [a,b] x Q: lim Yjit + h.^-Yjfru,) doesnQt^.A
Since both {Yj(t), a < t < b}, j — 1,2, have all sample paths continuous,
these two processes are also measurable (see Exercise 9.15) and, therefore, A\ and
Ai are (product) measurable. Since almost all the realizations of both {Yj(t), a <
t < b}, j'■ = 1,2, are absolutely continuous, we conclude that for almost every
w £ Q, the Lebesgue measure of the section of Aj with w fixed is 0 for both
j = 1,2 and, hence, by Fubini's theorem,
(LebxP)(A,-) = 0, j = 1,2.

526 MEASURABILITY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.7
Applying Fubini's theorem once again, we note that there is a Borel set To C [a, b]
of Lebesgue measure 0 such that for every t g To,
Now let
P[ lim -ii [ ii-i exists) = 1, j = 1,2. 11.7.4)
Di(t,u)
0 otherwise.
(11.7.5)
{Di(t)t a < t < 6} and {£>2(£), a < t < b} are measurable stochastic processes
and Relation (11.7.4) implies that they are versions of each other. It is also clear
that, in the notation of (11.7.1), for every w G Q, D\ (t, u) = Y\ (t, u>) for almost
every a < t < b, and so D\(-,u) G £^[0,6] for every u> G £1. Since JD] and
D2 are versions of each other, £>2(->w) S £p[a, 6] for almost every weQ, and,
therefore, apart from a null set, all realizations of {Y^), a <t <b} are, indeed,
in LP[a, b}. I
We now turn to a-stable processes with integral representation (10.1.1). Unless
stated otherwise, we assume in the sequel that the control measure m in the
integral representation is cr-finite. Note that a-finiteness is essentially implied by
Theorem 11.1.1 since a stochastic process with a version in ACp[a, b] always has
a measurable version (Exercise 9.15).
The following theorem is the main result of this section.
Theorem 11.7.4 Let {X(t), a < t < b} be an a-stable process 0 < a < 2,
with an integral representation (10.J.J). If a = 1, we assume in addition that the
random measure M is symmetric. Let p > 1. Then {X(t), a < t < 6} has a
version with all sample paths in ACp[a, b] if and only if
tj: [a,6]->R is in ACp[a,b], (11.7.6)
and there is a function f°: [a, b] x E —> K such that
(i) /°(t,x) = f(t,x)Tn-almost everywhere for any a<t<b, ..>
00 /°("jx) e ACp[a,b] for every x G E','and, moreover,
f°(t,x) = f{a,x) + / f°{s,x)ds, a<t<b,^
Ja
where
f: [a,b}xE->R

11.7 ABSOLUTE CONTINUITY OF STABLE PROCESSES 527
is a jointly measurable function satisfying
l-b
f U \f°{t,x)\am{dx)yadt<oo if l<p<a, (11.7.7)
/ [ \f°{t,x)\a[l+\og+A{t,x)]m{dx)du<ooifp = a, (11.7.8)
Ja JE
la JE
where
rb
A{t,x)- r tx
\f°(t,x)\" £ fE\fQ(u,v)\am(dv)du
fE\f°(t,v)\°m(dv)£\f°(u,x)\°du
and
f [f \f(t,x)\pdty Pm(dx)<oo if p>a>0. (11.7.9)
PROOF: Suppose firstly that {X(t), a < t < b} has a version with all sample
paths in ACp[a, b}. We start with the case when the random measure M is SaS
and 7] = 0 in (10.1.1). Since {X(t), a <t < b} is sample continuous, it can be
written in the form
X{t)= [ h(t,x)M(dx)3i.s., a<t<b, (11.7.10)
Je,
where E\ is a measurable subset of E such that the control measure m is ^-finite
on E\, and where h: [a, b] x E\ —* R satisfies:
(a) For every x e E\,
the function h(-,x): [a,b\ -* R is continuous, (11.7.11)
(b) for every t € [a, b], m(x € E{: h{t, x) ^ f(t, x)) = 0,
(c) for every t € [a, b],
m(ie£\£,:/(«,i)jt0). (11.7.12)
(See the discussion after Corollary 10.3.2.)
Let A be a probability measure on (Ei, £ n E\) equivalent to the control measure
m on this space. Then, by Theorem 3.10.1,
CXI
Z(t) = daY,^7i/aKt, Vi)s(Vi)l/a, a < t < b,
i=l

528 MEASURABILITY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.7
is a version of {X(t), a < t < b}, where g = dm/dX and, as usual, da is a
constant, and the three independent series of random variables are a Rademacher
sequence {ei,€2,...}, a sequence of arrival times of a Poisson process with
unit arrival rate {T\, I"^,.. ■} and a sequence of i.i.d. E\ -valued random variables
{V\, Vi,...} with common law A. Replacing £j by -£j fori = 2,3,... in the series
defining {Z(t), a < t < 6}, we obtain another version of {X(t), a < t < b},
say {2{t), a<t < b}. Both {Z(t), a < t < 6} and {Z(t), a < t < b} have
versions with all sample paths in ACp[a, b), and so does their sum,
Z{t) + Z(t) = 2dazxr-x'ag{Vxyiah{t, V{), a<t<b,
by Proposition 11.7.3. It follows from (11.7.11) and Proposition 11.7.3 that there
is a measurable set E2 Q E\ with m(£i\£2) = \{Ei\E2) = 0 such that
the function h{-, x): [a, 6] -♦ R is in ACp{a, b] Vx £ E2. (11.7.13)
Since (11.7.11)—<11.7.13) hold with jBi replaced by Ei, we will assume, for
notational ease, that E2 = E\.
Let {Y{t), a < t < b} be a version of {X(t), a<t<b} satisfying (11.7.2),
chosen in such a way that X(t) = Y(t) a.s. for every t £ [a, b]. It follows from
Proposition 11.7.3 that there is a Borel subset To of [a, b] with Leb(To) = 0 such
that for every t <£ To, D(t) — Y'(t) a.s. Therefore the stochastic process
f D{t) ifigTo,
D(t) = \
{ 0 ifteTo,
is a measurable SaS process. To determine its integral representation, observe
that for every to £ To,
D{t)= f h{t,x)M(dx) a.s., (11.7.14)
Jb,
where
h{tr)=^Ht + 6t'l~h{tr) inL"(E,,m).
6-+00 0
Note that according to Theorem 11.1.1, the function h: [a, b] x E\ -+ R (defined
= 0 for t € T0) can be chosen to be (jointly) measurable. Also note that for any
t £ To, there is a sequence 6n —> 0 such that
Un \ v h(t + 6n,x)-h{t,x)
h{t,x)= hm
n—00 0n
for m-almost every x € E\. In view of (11.7.13), there is a measurable set
£3 Q E\ with m(Ei\E3) = 0 such that for every x 6 £3, H'> x): (a? 61 ~* R is

11.7 ABSOLUTE CONTINUITY OF STABLE PROCESSES 529
in Lp{a, b], and
h(t,x) = h(a,x) + / h(s,x)ds, a<t<b.
J a
By applying Theorem 11.3.2 to {£>(£), a<t <b}, which has all its sample paths
in Lp[a, b] and setting
h(t,x) ifx&Ej,
0 ifx££3
h(t,x) ifxeEi,
and
f(t,x) =
0 ifx£E3,
we conclude the proof of the necessity part of the theorem in the SaS case.
The general case follows easily. If {Xi(t), a < t < b}, i = 1,2, are
independent copies of {X{t), a<t <b}, then {^i (t) - X2(i), a < t < 6} is a
SaS stochastic process with a version in ACp[a, b] and integral representation
Xi(t) - X2(t) = 21/a [ f{t,x)M(dx), a<t<b,
Jb
where M is a SaS random measure on (E, £) with control measure m. The
necessity part of the theorem (except (11.7.6)) follows from the SaS case
considered above. Relation (11.7.6) follows from the sufficiency part of the theorem,
which we shall now establish.
To prove sufficiency, suppose that (11.7.6) and conditions (i) and (ii) of the
theorem hold. Condition (ii) implies that there is a Borel set T0 C [a, b] of
Lebesgue measure 0 such that for every t £ T0, f°{t, ■) 6 La{m). Define
f JEf°(t,x)M(dx) ifi^To,
A(*) = <
[ 0 ifteTo.
Then {Di(t), a < t < b} is an a-stable process, and it follows from Theorem
11.3.2 that {£>i(i), a < t < b} has a version {D2(t), a < t < b} with all
sample paths in Lv[a, b). We know from Sections 11.1 and 11.2 that the version
{D2(t), a <t <b} can be chosen such that
JEf°{t,x)M{dx), iftgTo,
D2(t)
0 if t € To

530 MEASURABILITY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.7
a.s. for every t e [a, b]. Now let
*(*) = / f°{a,x)M{dx)+ [ D2(s)ds, a<t<b.
JE Ja
Theorem 11.4.1 implies that {X(t), a < t < b} is a well-defined stochastic
process and for any t € [a, b],
*{t) = f[f(a,x)+ f f>(s,x)ds^M(dx)
= f f°(t,x)M(dx)
JE
a.s. Obviously, {X{t), a < t < b} has all sample paths in ACp[a, b) and therefore
{X(t) + r](t), a < t < 6} is a version of {X(t), a < t < b} with all sample
paths in ACp[a, b}. This completes the proof of the theorem. I
The result of Theorem 11.7.4 extends to the complex-valued case, as the
following corollary indicates.
Corollary 11.7.5 Let {X(t), a < t < 6} be a complex-valued SaS process
given in the form
X(t) = / f(t,x)M(dx), a<t<b,
JE
where M is a complex-valued SaS random measure on (E, £) with control
measure m and circular control measure k, and f(t, ■) € La(m)foranyt E [a,b).
Letp> 1. Then {X(t), a<t <b} has a version with sample paths in ACp\a, b]
if and only if there is a function f°: [a, b] x E —> C such that (i) and (ii) of
Theorem 11.7.4 hold.
PROOF: Let X^t) = Re X(t), X2{t) = Im X(t), a < t < b. Then
{X\(t), a <t < 6} and {X2(t), a < t < b} are real-valued SaS processes, and
{X(t), a < t < b} has a version with sample paths in ACp[a, b] if and only if
both {Xi(t), a<t<b} and {^(i), a < t < b} do. It follows from Proposition
6.2.1 that {Xi{t), a < t < b} and {X2{t), a < t < b} have the following
integral representations (in the sense of joint finite-dimensional distributions):
J5sr,(t)= f (sjM(t,x)-s2f{2)(t,x))M{ds,dx), a<t<b, (11.7.15)
X2{t)= f (slf(-2)(t,x) + s2f(-l\t,x))M(ds,dx), a<t<b,
JExS2 V '

11.7 ABSOLUTE CONTINUITY OF STABLE PROCESSES 531
where f^(t,x) = Re f(t,x), f{2\t,x) = Im f(t,x), and M is a (real) SaS
random measure on (E x S2, £ x B) with control measure k. In the sequel,
we suppose, without loss of generality, that k (and therefore m) is a probability
measure.
Suppose firstly that {X(t), a < t < b} has a version with all sample paths
in ACp[a,b]. Then, both {Xi(t), a < t < b} and {X2{t), a < t < b) so do.
Applying Theorem 11.7.4 to the processes {X\(t), a < t < b} and {X2(t), a <
t < b), we conclude that there are functions /°W; [a, b] x E —» R, i = 1,2,
such that /°M(£,x) = f^(t,x) m-almost everywhere for every i = 1,2 and
a<*<6,/°W(-,x) €^Cp[a,6]foreveryi= 1,2and x € £, and
foii)(t,x) = fW{a,x) + / f0{i)(s,x)ds, i = l,2,a<t<b,
J a
for jointly measurable functions /0(i): [a,6]xB-»R,i = l,2. Moreover, the
functions /°W are such that the 5q5 processes
£>,(*) = /" (sj°w(t,x) - s2f°{2}(t,x))M(ds,dx), a<t<b,
and
D2(t)= f (Slf0M(t,x)+s2f°W(t,x))M(ds,dx), a<t<b,
have almost all sample paths in Lp[a, b\. Let (t\, t2, ...) be a sequence of i.i.d.
random variables uniformly distributed on (0,6) and {{Z\, Ai), (Z2, A2),-..) be
a sequence of i.i.d. E x ^-valued random vectors with common distribution A;,
independent of [t\ , t2, ...). By (11.3.4),
sup sup n xlpj 1//q
n>lj>\
sup supn l'pj x'a
n>\j>\
where Aj = {Af,Af), j = 1,2, Relations (11.7.16) and (11.7.17) imply
sup sup n-VPj-2/- (/0<Vn,^)2 + /°(2)(Tn,^)2) < °° a-s->
n>lj>l V '
< 00 a.s.,
(11.7.16)
< 00 a.s.,
(11.7.17)
so that
sup supn-^pj-1/Q|/0(i)(rn, Z,-)| < 00 a.s. for i = 1,2.
n>lj>l

532 MEASURABILITY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.7
The SaS processes
' Wi(t) = [ f°^(t,x)M(dx), a < t < b, i = 1,2,
J E
have, therefore, almost all their sample paths in Lp[a, b] (see the remark following
Theorem 11.3.2). By Theorem 11.3.2 both f°W, i = 1,2, satisfy (11.7.7)-
(11.7.9). Then setting /° = f°W + if°W completes the argument.
The second part of the statement also follows from the corresponding real-
valued case: just reverse the preceding steps. I
Example 11.7.6 Let {Z(t), a < t < b} be a stationary harmonizable SaS
process (see (6.5.1)) and p > 1. We apply Corollary 11.7.5 to find out when
this process has a version with all sample paths in ACp[a, b]. Clearly, the only
candidate for f is / itself, i.e., f°(t,x) = eitx, a < t < b, x 6 R. Then
f°(t,x) = ixeitx, and \f°(t,x)\ = |x|. A trivial check of (11.7.7H11-7.9)
shows that these conditions hold if and only if
/oo
\x\am(dx) < oo. (11.7.18)
-oo
Therefore (11.7.18) is a necessary and sufficient condition for a stationary
harmonizable process to have a version with all sample paths in ACp\a, b]. Note that
this.condition does not depend on p !
Now let X(t) —ReZ(t), a < t < 6, be a real harmonizable SaS process.
Applying Theorem 11.7.4 to the integral representation (11.7.15), we see that for
every p > 1, (11.7.18) is also the necessary and sufficient condition for a real
harmonizable SaS process to have a version with all sample paths in ACp[a, b}.
Example 11.7.7 Let {X(t), — oo < t < oo} be a stationary a-stable moving
average
/oo
f{t - x)M(dx), -oo < t < oo,
-oo
where / € La{—oo,+oo) and M is an a-stable random measure on (M,B)
with Lebesgue control measure and constant skewness intensity /3 (we assume
0 = 0 if a = 1). The discussion in Example 10.4.5 shows that if X is to
be sample continuous, let alone have an absolutely continuous version, / must
have a continuous modification. We assume, therefore, from the outset that / is
continuous.
We now apply Theorem 11.7.4 to find conditions for the moving average
process to have a version in ACp[a, b], p > 1. The stationarity of the process
implies that if such a version exists for some [a,b], then there is a version in
ACp[a, b] for any a < b, so we may as well set a — 0 and 6=1. Since the only

11.8 EXERCISES 533
candidate for f° in Theorem 11.7.4 is / itself, we conclude that the necessary and
sufficient conditions for a stationary a-stable moving average to have a version in
ACp[a,b]aie:
(i) / is absolutely continuous,
(ii)
/•OO
|/'(x)|Qdx<ooif 1 <p<a,
f
r\ /•CO
/ / i/'wr
JO J -oo
r
J — a
1 + 10g+
\f'(x)\"
Fa(x + t)\
Fp(x)a/pdx < oo if p > a,
dxdt < oo if a = p,
where
Fp(x) = £ \f(t)\pdt.
In particular, the moving average
/oo
e~lt-xlM{dx), -oo < t < oo,
■oo
has a version in ACp[a, b] for any 0 < a < 2, p > 1.
Example 11.7.8 A simple application of Theorem 11.7.4 shows that the well-
balanced linear fractional stable motion (3.6.4) with 1 < a < 2 and 1/a < H < 1
has no versions in ACp[a, b]. It is, however, sample continuous as we will see in
the next chapter.
11.8 Exercises
Exercise 11.1 (i) Prove the Khinchine inequality: for any p > 0, any n =
1,2,...,
i=l i=l
where a*,» = 1,..., n are real numbers, £i, i = 1,..., n are Rademacher random
variables and C(p) is a finite constant depending only on p.
Hint: Prove it for p = 2m first,
(ii) Prove the reverse inequality
n n
E|X>£* >c(p)(£a?)
i=i »=i
P/2

534 MEASURABILITY, INTEGRABILITY AND ABSOLUTE CONTINUITY 11.8
for p > 2, where c(p) is a positive constant depending only on p.
(iii) Prove that for any random variable Z with a finite fourth moment and for any
(mi>A))'/2>g^|.
Hint: Write EZ2 = E(Z2l(\Z\ > A) + Z2l{\Z\ < A)) and use the Holder
inequality.
(iv) Prove that for any p e (0,1),
p(|f:aiei|>p(X:a?),/2)>C(l-p2),
1=1 1=1
where C is a positive constant.
(v) Use (iv) to extend (iii) to all p > 0 and to show that the series Y^Zi a*e'
converges a.s. if and only if it converges in Lp for every p > 0.
(vi) Suppose that the series ]T)~, a^ converges. Then, for every p > 0, there
are positive constants c(p) and C{p) such that
c(p)(E0 <4zH <c(p)(:e«?) •
Exercise 11.2 Let{Xj(£), £ € T},z = 1,2, be independent measurable copies of
a strictly a-stable process, ct^l, defined on a cr-finite measure space (T, T, y).
Prove that
\X{{t)\pu{dt) < oo a.s.
I
if and only if
r \Y(t)\pv{dt) < co a.s.,
I
IT
where Y(t) = 2'xla{X^ (t) - X2(i)), t 6 T.
Exercise 11.3 Let (T, r, v) be a cr-finite measure space and let X — {X{t), t S
T} and {Y = Y(t), t S T} be two measurable a-stable processes with all sample
paths in LP(T, t, i/) and such that
{X{t), teT}± {Y(t), t e T}
(equality of the finite-dimensional distributions).
Show that
/ \X(t)\pu(dt) £ [ \Y{t)\pu{dt)
Jt jt

11.8
EXERCISES
535
and, moreover,
X = Y
(equality of the distributions of two random elements in LP (T, r, u)).
Hint: Proceed as in the beginning of Section 11.4 using the law of large numbers
to establish convergence to JT \X(t)\pu(dt) and fT \Y(t)\pi/{dt). To prove that
X = Y in LP, one needs to show that the probabilities that X and Y belong to
the same open ball in LP are equal, i.e., P(\\X - z\\p < A) = P(||V - z\\p < A).
This, however, follows from the first part of the exercise with X and Y replaced
by X — z and Y — z, respectively.
Exercise 11.4 Verify (11.6.5).
Exercise 11.5 Verify the properties of the functions g^.' (x) and g^. (x) described
in the proof of Theorem 11.6.1.

Chapter 12
Boundedness and continuity
via metric entropy
In Chapter 10, we obtained necessary conditions for sample boundedness and
sample continuity of stable processes of index 0 < a < 2 in terms of the kernel in
the integral representation. We also observed that these conditions are sufficient
when 0 < a < 1 but not when 1 < a < 2 (Example 10.4.1 and Exercise
10.7). No necessary and sufficient conditions are presently known in the case
1 < a < 2. In this chapter, we use "metric entropy" to obtain sufficient conditions
and additional necessary conditions for sample boundedness and continuity in the
case 1 < a < 2.
Observe that the results of Section 11.7 also provide sufficient conditions
for sample path boundedness and continuity when 1 < a < 2. Indeed,
Theorem 11.7.4 gives necessary and sufficient conditions for an a-stable process
{X(t), a < t < b} to have a version with absolutely continuous sample paths;
since any such version has continuous sample paths, sufficient conditions for
sample continuity follow. There are, however, a-stable processes that are sample
continuous but do not have versions with absolutely continuous sample paths, for
example, the linear fractional stable motion (Examples 11.7.8 and 12.2.3).
Metric entropy offers an alternative to integral representation techniques. It
has been extensively used in the context of Gaussian processes. Because a-stable
processes are closely related to Gaussian processes through conditioning, it is
natural to check whether the techniques used in the Gaussian case apply. They
do, but the "gap" between the necessary and the sufficient conditions for sample
, regularity is much greater in the a-stable case than in the Gaussian case.
In Section 12.1 we review the Gaussian results and set the scene for the
discussion of the a-stable case. Sufficient and necessary conditions in the a-

538 BOUNDEDNESS AND CONTINUITY VIA METRIC ENTROPY 12.1
stable, 1 < a < 2, case, are given, respectively, in Sections 12.2 and 12.3.
Finally, in Section 12.4 we collect a number of facts, scattered throughout the
book, which are related to boundedness and continuity of a-stable il-self-similar
processes.
12.1 Metric entropy
Consider a zero mean Gaussian process X = {X(t), t e T}. The process X
generates the following pseudometric dx on T:
dx(t,s) = (E(X(t)-X(s))2y/2, t,s€T. (12.1.1)
For any given e > 0, let N(e) denote the smallest number of open d^-balls of
radius e needed to cover the parameter set T. The number N(e), which is called the
covering number, may be infinite. It is, of course, finite for all e > 0 if and only if
(T, dx) has a compact closure. In this case, N(e) = 1 when e > supt dx (t, s).
Since N(e) increases as e decreases, it is the behavior of N(e) around e = 0 that
is of interest.
Definition 12.1.1 The function H: (0,oo) -+ [0,oo] defined by H(e) = logiV(e)
is called the metric entropy (function) of the process {X(t), t € T}.
A fundamental theorem of Dudley (1973) states that if
oo
H(e)l/2de < oo, (12.1.2)
then the Gaussian process {X(t), t e-T} is both sample bounded on T and sample
continuous in the metric dx- If T is endowed with a topology that makes dx a
continuous function (i.e., as n —* oo, tn —► t =4> dx{tn,t) —► 0; equivalently
{X(i), t € T} is continuous in probability), then (12.1.2) also implies that
{X(t), t € T} is sample continuous in the topology of T. Moreover, if (12.1.2)
holds, then a separable version {Y(t), t € T} of {X(t), t 6 T} satisfies
/•OO
EsupY(t)<K H(e)1/2de (12.1.3)
ter Jo
where K is a universal constant.
Metric entropy also provides necessary conditions for sample regularity of
Gaussian processes. Namely, if {X(t), t € T} is sample bounded, then
svpeH(e)l'2<oo, (12.1.4)
£>0
I

12.1
METRIC ENTROPY
539
and if (T, dx) is relatively compact and {X(t), t € T} is sample continuous in
the metric dx, then
lirne#(e)1/2 = 0. (12.1.5)
Moreover, any separable version {Y(t), t e T} of {X(t), i € T} satisfies
£sup Y{t) > K~x supeH(e)^2 (12.1.6)
where K is a universal constant.
There does not appear to be a big "gap" between the necessary conditions
(12.1.4) and (12.1.5) for sample boundedness and continuity and the sufficient
condition (12.1.2). In fact, although (12.1.2) is not necessary for sample continuity,
it becomes so when (T, dx) is relatively compact, T is a group and {X(t), t € T}
is a stationary process (so that dx is a translation invariant metric on T). For a
proof of these results see Dudley (1973) and Femique (1975).
How useful is the metric entropy approach in the case of a-stable processes,
1 < a < 2? It turns out that metric entropy does provide information on sample
boundedness and continuity, but the gap between the necessary conditions and
the sufficient conditions widens. This can be attributed to the fact that, unlike
the Gaussian case, a single number, "the distance" between two points X(t) and
X(s), does not tell much about their joint distribution.
Let {X(t), t e T} be a SaS process, 1 < a < 2. For s,t € T, define
dx{s,t) as the scale parameter of the SaS random variable X(t) - X(s). Note
that
dx(s,t) = \\X(t)-X{s)\\a
is also the covariation norm of X(t) — X(s); see Section 2.8 and Exercises 2.22,
2.23 and 2.24. Let N(e) and H(e) be defined as above (but now, of course, in
terms of the new metric dx)-
We shall firstly try to obtain sufficient conditions for sample path regularity
for {X(t). t € T}. It is convenient, however, to use a slightly more general
framework than one specific to a-stable processes.
Let T be a separable metric space and let {X(t), t € T} be a strongly
separable stochastic process defined on a probability space (Q,:F,P). Consider
L°(C1, T, P), the space of all random variables on (Q, T, P), and let || • || be an
extended-valued continuous semi-norm on L0(O, T, P), i.e.,
1. ||X|| e [0,oo] for all X e L°(£l,T,P),
2. ||JC||=0iffX = 0a.s.,
3. ||X + Y|| < ||X|| + ||Y|| for any X,Y 6 L°(Q,^,P),

540 BOUNDEDNESS AND CONTINUITY VIA METRIC ENTROPY 12.1
4. ||*n|| T 11*11 as n-*<x> for any *,Xn G L0(£i,F,P), n= 1,2
such that \Xn\ | \X\ a.s. as n —* oo.
(The linearity condition \\aX\\ = \a\ \\X\\ is not imposed.) Define also
L" = {XeL°(Q,T,P): pC||<oo}.
The following theorem is a slight generalization of Theorems 11.2 and 11.6
of Ledoux and Talagrand (1991) whose proof we follow.
Theorem 12.1.2 Suppose that X(t) € iJHI for any t £ T. Let dbea (pseudo)-
metric on T and <f>be a concave non-decreasing function on (0, oo) such that, for
any A € T and s,t € T,
\\(X(s)-X(t))\A\\<d(s,t)P(A)4>(l/P(A)). (12.1.7)
Then for any r) > 0,
/■166
|| sup \X(s)-X{t)\\\ <ri<j>(N{6)2) + 32 / <t>(N(e))de (12.1.8)
s,t€T, d(s,t)<7] JO
for any 0 < 5 < D where
D — sup d(s,t)
s,teT
is the diameter of(T, d), andN(e) is the minimal number of open d-balls of radius
e needed to cover T.
We start with a lemma.
Lemma 12.1.3 Let X\, X2,..., Xn be in L"" such that for any A 6 T and
i=l,...,N,
\\\Xi\lA\\<P(A)(l>(l/P(A)). (12.1.9)
Then, for every A € T,
|| .jnaxJXil^ll < P(A)4>(N/P(A)).
Proof: Let Ai = {w 6 ,4: |Xi| > maxj=1>...,w |X,|}, i = 1,..., N, and let
Bi = MU5=i 4p t = 1,... ,n. Then, by (12.1.9),
|| . max MUH = HEI^IIbJI < Ol^MI
i=l,...,JV t—: . \
J=I J=l
<
" P(B,) 1
^wp^hww)).

12.1
METRIC ENTROPY
541
using the concavity of <j>. 1
Proof of Theorem 12.1.2: Let T* = {tut2,...} be a strong separant for
{X(t), t e T} and T* = {ti,h,...,tn}, n = 1,2,... . We prove that for
every n— 1,2,... and rj > 0,
/•166
|| max |X(S) - X(t)| || < t,4>{N{S)2) + 32 / 4>(N(e))de (12.1.10)
for every 0 < 6 < D. Relation (12.1.8) then follows because, as n —> oo
max |X(s) - X(t)| T sup \X{s)-X(t)\= sup |X(a)-X(i)|.
s'teTn «.t£T- s,t£T
<*(».0<1 c((s.t)<>J <*(•».*)■<«?
Fix an n > 1 and consider T£. For each m = ..., —1,0,1,..., let Sm
be a subset of T* of minimal cardinality such that the open balls of radius 2~~m
with centers in Sm cover T*. Denoting the cardinality by #, we have #(Sm) <
iV(2-(m+')) (see Exercise 12.2). For each t € T*, fix an sm(t) e. Sm such that
d(t,sm(t)).< 2~m. Since T£ is a finite set, there is anm such that d(t,sm{t)) = 0
for every t € T*. Let m0 be the smallest such m. For every m <mo and tsT^,
mi
*(*)= £ (X(fc(i))-*(**_,(*)))+X(Mt)), (12.1.11)
i=m+l
where fori < m0, fcj(i) = Si(si+i(- ■ ■ (s^M)...)). Then
max|X(t)-X(fcm(i))| < Y max\X(ki(t))-X{ki-i(t))\. (12.1.12)
teT- i=m+iter"
Observe that for any t € T* andi < m0,d{ki{t), fc»_i(t)) < 2-(i_1>. Moreover,
the collection {X(ki{t)) - X(fci_i(t)), t € T*} consists of at most #{Si) <
N(2~(i+l)) distinct random variables. It follows now from (12.1.7) and Lemma
12.1.3 that for any measurable set Asf,
||lAmax|Z(fci(t))-X(fci_1(i))| || < 2-(i-Vp(A)4>(N(2-(i+l>)/P(A)),
ii tl-T.
and so, using (12.1.12),
mo
||U max \X(t) - X(km(t))\ \\ < P(A) £ 2-^U(N(2^+^)/P(A)).
t€T" i=m+t
In particular, for A = Q. and 5 > 0 such that 2-(m+2) < <5 < 2-(m+1>,
i-16<5
ix \X(t) - X(km(t))\ || < 8 / 0(JV(e))de. (12.1.13)
r* Jo
max
iter-

542 BOUNDEDNESS AND CONTINUITY VIA METRIC ENTROPY 12.2
Keeping m < m0 fixed, define for any 77 > 0, Um(ji) = {(x^v) € Sm x
Sm: there aret € T*, s e 7^ with d(s,i) < 77 such that km(t) = x, km(s) = y}.
Obviously, #(C/m(r?)) < JV(2-(m+»)2 < JV(6)2. For any (x,y) 6 Um{rj),
fix a t(x,y) e T* and s(x,y) G T* such that d(t(x,y), s(i,y)) < 77 and
fcm(*(a;,y)) = 2, fcm(s(x,y)) = y- Applying (12.1.7) and Lemma 12.1.3, we
conclude that
|| max \X(t(x,y))-X(s(x,y))\\\<r,<P(N(6)2). (12.1.14)
Take now any t 6 T*, s € T£ such that d{t, s) < 77. Clearly, (fcm(t), fcm(s)) €
L/m (77) and thus
|X(s)-X(t)|
< \X(s) - X(km(s))\ + \X(km{s)) - X(s(km(t), km(s)))\
+ \X(s(km(t), km(s))) - X{t(km(t), km(s)))\
+ \X(t(km(t), km(s))) - X(fcm(t))| + \X(km(t)) - X(t)\
<4max|X(t)-X(fcm(t))|+ max \X(t(x,y)) - X(s{x,y))\.
te~n (x,v)ec/m(i?)
Using (12.1.13) and (12.1.14), we obtain
fr-16,5
0
Jr\6S
' 4>(N{e))de,
0
d(«,t)<rl
which proves (12.1.10) and completes the proof. I
12.2 Sufficient conditions in the case 1 < a < 2
The following theorem gives sufficient conditions for sample boundedness and
continuity of SaS processes, 1 < a < 2.
Theorem 12.2.1 Let {X(t), t € T} be a SaS process, 1 < a < 2. Let
dx(s,t) = \\X{s) - X(t)\\a, s ST, teT.be the (pseudo) metric generated by
the process. Let D = sups teTdx(s,t) be the dx-diameter ofT. If
I
D
N{e)l'ade < oo and 1 < a < 2 (12.2.1)
o
or if
rD
| log e\N{e)de < oo and a = 1, (12.2.2)
/
Jo

12.2 SUFFICIENT CONDITIONS IN THE CASE 1 < a < 2 543
then {X(t), t 6 T} is both sample bounded and sample continuous (in the metric
dx).
Moreover, if{Y(t), t £ T} is a strong separable version of{X(t), t 6 T}
defined on a probability space (Qj, T\, Pi) and 1 < a < 2, then, for any r\ > 0,
Ex sup \Y(s)-Y(t)\<T]-^—raN(8)2/a + 32-^-ra [ N(e)^ade
»,(er a — 1 a — 1 Jn
(12.2.3)
foranyQ <6<D. Here,r% = supA>0 AQP(|X0| > A),w/iereX0 ~ Sa(l,0,0).
Proof: Let {Y(t), t € T} be a strongly separable version of {X(t), t € T}
defined on a probability space (Qi, T\, Pi).
We start with the case 1 < a < 2. For X € L°(Qi,.Fi.Pi), set ||A"|| =
£i|X|, d(s,t) = dx(s,t), s,t€T, and take <£(x) = a'rQxlla, where a' is the
conjugate of a (i.e., 1/a + 1/a' = 1).
Observe that for any i€fi, s, t € T, and any e > 0
\\iA(Y(s)-Y(t))\\ = £,|u|y(s)-y(t)|
/*00
= / p,({|y(s)-y(t)|>A}nA)dA
Jo
/»€ /»00
Jo is
/oo
p,(|y(s)-y(i)|>A)dA
/oo
d(s,i)QA-Qr^dA
= fPi^+rSd^^^a-l)-^-^1
= e[Pl(A)+ry(s,t)a(a-irle-a}.
Choosing e = d(s,i)raPt(>l)_1/Q, we see that Condition (12.1.7) of Theorem
12.1.2 holds with 4>{x) = ^rxrax^a. Then (12.2.3) holds as well and we have,
under the assumption (12.2.1),
limiB, sup \Y{s)-Y{t)\ = 0,
1—»0 «,t€T
dx(s,t)<T
so that
lim sup |Y(s)-Y(t)|=0 a.s.
I)—*0 a.tgT
dx(a,l)<r,
This proves that all sample paths of {Y (i) ,t€T} are continuous in the metric dx
on a set of Pi -measure 1. Since (T, dx) is relatively compact under the assumption

544 BOUNDEDNESS AND CONTINUITY VIA METRIC ENTROPY 12.2
(12.2.1), all sample paths of {Y(t), t e T} on that set are also bounded. Therefore,
{X(t), t e T) is both sample bounded and sample continuous in the metric dx.
We now consider the case a = 1. We set this time \\X\\ = E\ min(l, \X\) for
X € L°(Qi,.Fi,Pi). Assumption (12.2.2) ensures that I? < oo. Choose ana > 0
large enough so that f(x) = xlog(a + ^) is a positive non-decreasing concave
function on (0,D], define /(0) = 0 and set d(s,t) = f(dx(s,t)), s,t e T.
Observe that d(s, t) is a (pseudo)-metric on T and that d(s, t) —> 0 if and only if
dx(s, t) —> 0. Choose also a finite positive constant c\ large enough so that
£min(l,|Jf|)<c,<7log(a + -),
for any S\(cr, 0,0) random variable X with 0 < a < D. Finally, let <p(x) = c2x.
Then Condition (12.1.7) of Theorem 12.1.2 holds if c2 > 0 is large enough, and
we have, for any 77 > 0 and 0 < 6 < f{D),
/ \ - ■ ri6S ~
Eminfl, sup \Y(s) - Y(t)\) < c2t1N{6)'1 + 32c2 / N(e)de (12.2.4)
\ «,t€T / J0
d(a,t)<>)
where N(e) is the minimal number of open d-balls of radius e required to cover
T.
Obviously, for any 0 < e < f{D),
N(e) = N(/-'(c))
< Ar(c3e/log(a+i))
for some C3 € (0,00). Substituting this into (12.2.4) and changing the variable of
the integration, we conclude
j^minfi, sup |y(s)-y(t)|)
d(s,t)<i)
< c2r]N( c36/ log (a + -J J +C4/ ^l + |logu|J7V(u)du,
(12.2.5)
where 0 < C4 < 00. It follows now from (12.2.2) that
lim E, minf 1, sup \Y(s) - Y(t)\) = 0,
tj—»0 V ».ter /
d(»,t)<T)
which implies, as before,
lim sup \Y(s) - Y(t)\ = 0 a.s.
d(»,t)<1

12.2 SUFFICIENT CONDITIONS IN THE CASE 1 < a < 2 545
Then, on a set of Pi-measure 1, all sample paths of {Y(t), t € T} are continuous
in the metric d and, therefore, in the metric dx as well. Hence {X(t), t € T} is
both sample bounded and sample continuous in the metric dx- 1
An analogous result holds in a more general case:
Corollary 12.2.2 Let {X(t), t € T} be a strictly a-stable process, 1 < a < 2,
and let dx{s,t) be the scale parameter of {X(t) — X(s), s,t £ T}. Suppose
that (12.2.1) holds. Then {X(t), t 6 T] is both sample bounded and sample
continuous in the metric dx- Moreover, if{Y(t), t £ T} is a strongly separable
version of {X(t), t € T} defined on a probability space (Cli,jF\,Pi), then for
any r] > 0, Relation (12.2.3) holds, provided we multiply its right-hand side by
21/Qa'', where a' is the conjugate of a (i.e., \/a + \/a' = 1).
PROOF: Let Z(t) = 2-'/Q(X1(t) - X2(t)), t e T, where {Xt{t), t G T},
i = 1,2, are independent copies of {X(t), t 6 T}. Then {Z(t), t € T} is a
SaS process generating a metric dz(t, s) = dx{t, s) for every t,s € T. Fix any
countable subset C of T. Now (12.2.3) implies
E sup \Z{t) - Z{s)\ > 2-'/a(l - l/a)E sup \X(t) - X(s)\.
t,sec t.sgc
Thus, if {Y(t), t 6 T} is a strongly separable version of {X(t), t 6 T} defined
on a probability space (Qi, ^"i, Pi), then, by (12.2.3)
Ei sup \Y(t)-Y(s)\
t,s£T
dx{t,s)<-n
l_i) J^rQr]N(S)V° + 32 N{ey'ade . (12.2.6)
a / a — 1 L J0 J
Under assumption (12.2.1), Relation (12.2.6) implies, as before, both the sample
boundedness and sample continuity in the metric dx of the a-stable process
{X(t),teT}. I
Remark. In the case 1 < a < 2, Relation (12.2.3) of Theorem 12.2.1 provides
an upper bound on the "size" of the oscillations of the process {X(t), t € T}.
An upper bound on the "size" of the oscillations in the case a = 1 is given in the
proof of the theorem (see (12.2.5)).
Example 12.2.3 We have seen in Example 10.2.5 that a well-balanced linear
fractional a-stable motion with H < 1/q is not sample bounded on any interval
of positive length. Suppose, now, that 1/a < H < 1, which is possible only
when 1 < a < 2. We will show that in this case, {X{t), -co < t < co} is

546 BOUNDEDNESS AND CONTINUITY VIA METRIC ENTROPY 12.3
sample continuous. According to Proposition 9.3.5, it is enough to show that
for any -co < a < b < oo, {X(t), a < t < b} is sample continuous. Fix
-co < a < b < co and s,t € [a, b}. By the iif-self-similarity of the process and
the stationarity of its increments (see Exercise 3.9) we have E\X{t) - X(s)\p =
Ca<p\t — s\Hp for any 0 < p < a, and hence the metric generated by the process
is of the form
dx{t,s) = ca\t - s\H, a<s,t<b.
Apply Theorem 12.2.1. Since
N(e) < const.£-'/w, 0 < e < D = ca(b - a)H,
the assumption l/a < H ensures that Condition (12.2.1) of the theorem holds.
Therefore {X(t), a <t < b} is sample continuous.
As we will see in the sequel, there is a big gap between the sufficient conditions
for sample path boundedness and continuity given in Theorem 12.2.1 and the
corresponding necessary conditions. But in some sense, both type of conditions
are the best possible. As far as the sufficient conditions are concerned, this
can be seen, for example, by considering the log-fractional stable motion of
Example 3.6.6 (1 < a < 2). For every —co < a < b < oo, this process
{X(t), a < t < b} is not a.s. bounded because the necessary conditions of
Theorem 10.2.3 fail. But this process is 1 /a-self-similar and has stationary
increments so dx(s,t) = ca\s-t\^a, a < s,t < b, and thus N(e) ~ const.e~1/0,
as e —> 0. The sufficient condition of Theorem 12.2.1 fails (as, of course, it must),
but barely (the integral J0 N(e)l^ade has a logarithmic divergence).
In fact, one can show the following:
For any non-increasing function h: (0, oo) —> (0, oo) with h(6) | oo as 6 [ 0,
there is a SaS process {X(t), 0 < t < 1}, 1 < a < 2, such that
/ N{e)l/ade < h(9) for every 8'€ (0,1)
Je
and {X(t), 0 < t < 1} is not sample bounded. (See Exercise 12.1.)
12.3 Necessary conditions in the case 1 < a < 2
We now turn to necessary conditions for sample path regularity. Here is the main
theorem.
Theorem 12.3.1 Let {X(t), t 6 T} be a SaS process, 1 < a < 2. Let
dx{s,i) = \\X{s) - X(t)\\a, s e T, t e T, be the (pseudo)-metric generated

12.3 NECESSARY CONDITIONS IN THE CASE 1 < a < 2 547
by the process. Then there is an absolute constant Ka € (0, oo) such that for a
separable version {Y(t), t G T) defined on a probability, space [Q.\,T\,P\),
Ei sup|y(t)| > Kae (logiV(e))1/"' (12.3.1)
if I < a < 2, where a' is the conjugate of a, and
TOI1/2'2
(^supTOI1/2)2 >/r,elog(l+logW(C)) (12.3.2)
ifa= 1.
In particular, if{X(t)j t € T} is sample bounded, then
if\ < a < 2 and
sup£(logiV(e))1A*' <oo (12.3.3)
e>0
supelog(l+logiV(£)) <oo (12.3.4)
£>0
if a — 1. Moreover, if(T,dx) is relatively compact and {X (t), t E T} is sample
continuous, then
lime(logJV(e))|/Q'=0 (12.3.5)
£—►0
if\ < a < 2 and
ifa= 1.
limelog(l+logAT(e)) =0 (12.3.6)
Remark. Relations (12.2.1), (12.3.3) and (12.3.5) are extensions to the case
1 < a < 2 of the corresponding Gaussian relations (12.1.2), (12.1.4) and (12.1.5)
because a' = a when a = 2. One can now clearly notice the "gap" between the
necessary conditions for sample path regularity given in Theorem 12.3.1 and the
sufficient conditions of Theorem 12.2.1. The gap in the case 1 < a < 2 is much
wider than in the Gaussian case a = 2.
Proof of Theorem 12.3.1: We will omit the proof of (12.3.2) and refer the
reader to Talagrand (1988) instead. For (12.3.1), fix any e > 0 and use Exercise
12.2 to conclude that there are points t\,...,t;v(e) in T such that for any i, j =
l,...,JV(e), i # j, dx(ti,tj) > e. Obviously, (12.3.1) will follow once we
show that for some absolute constant Ka
E max \X{ti)\>Kae (log N{e))i/a'. (12.3.7)
i=l,...,N(£)

548 . BOUNDEDNESS AND CONTINUITY VIA METRIC ENTROPY 12.3
Applying Proposition 3.11.1 to the SaS random vector {X(t\),.. -,X(tN^))),
we see that it can be represented as a probability mixture of zero mean Gaussian
vectors. Namely,
where the SaS random vectorZ = {Z\,...,ZN^) is defined on the product
of two probability spaces, (Qi x Q.i,T\ x F2, P\ x P2). and for each u^ e Q,
Z(wi, •) is a zero mean Gaussian vector. Let
<Wm) = (Wi(wi,W2) -^•(w„w2))2),/2,i>j € {l,...,iV(e)},
and note that for any real 0 and i, j € {1,..., N(e)}
e-\erdx(t,tjr = EexviiBiXitJ-Xfo))}
= fiexp^Zi-Z,-)}
= Ei(E2t\pie(Zi-Zj))
= EIe-ifl2<M,''*>\ (12.3.8)
This identifies dUt(i,j)2 as a positive a/2-stable random variable. For a fixed
u/i x Qi,letA(wi) = niinijg^^.^fe)}, i& du,(hj)- We immediately conclude
from (12.1.6) and Exercise 12.2 that
E2 . max \Z{\ > K-^-A^ilogNie))1/2,
i=l,...,iV(e) JL
so that
E maxu\Xi\ = B max \Zt\ > -^(logiVCe))1/2^^,). (12.3.9)
i=l,...,JV(c) t=l,...,iV(e) Z/V
By Markov's inequality, for any <5 > 0 and 8 > 0,
P(du,(i,j)<6) = p(e-^^,(^)2>e-Je262)
- e-l/292«2
= expji^-ierdx^.ti)01}.
Choosing 0 = ^-2/(2-0=)^^^^ ^a/p-o,^ we conciucie that for any 6 > 0
P(<U*J) < S) < exp{-i(Wx(ti>ti))-2a/(2-a)}-

12.3 NECESSARY CONDITIONS IN THE CASE 1 < a < 2 549
Using Exercise 12.3, we immediately conclude that for a finite positive constant
J5A(w,) > A;afn-log7V(e)-(2-Q)A2Q) min dx{ti,tj))
> ka(l+\ogN(e))-{2-am2a)e.
Substituting the last estimate into (12.3.9) yields (12.3.7).
Further, if {X(t), t € T} is sample bounded, then, by Proposition 10.2.1,
£supt6T|Y(t)| < oo,andnow(12.3.3)followsfrom(12.3.1). Similarly,(12.3.4)
follows from (12.3.2).
Let (T, dx) be relatively compact, and assume that {X(t), t € T} is sample
continuous with a = 1. It again follows from Proposition 10.2.1 that
limU, sup |y(i)-Y(s)|'/2 = 0.
Fix an 77 > 0 and choose a 5 > 0 such that £supd ,t s)<6 |Y(£) — Y(s)|'/2 < 77.
We can cover T with finitely many open dx-balls of radius 8,B\,..., Bm. Then
EX sup \Y(t)-Y(s)\^2<ri,
t,seBt
i = 1,... ,m. It follows from (12.3.2) that for each i — 1,... , mande > 0,
elog(l+logAri(e))<X-|r?I
where Ni(e) is the minimal number of open balls of radius e needed to cover the
set {(s, t): s G Bi, t 6 Bi] = Bi x B, in the metric
d((si,U), (s2,t2)) = \\(X(U) - X(Si)) - (X(t2) - X(s2))h.
If iVj(e) denotes the minimal number of open dx-balls of radius e needed to cover
Bi, then, clearly, Ni(e) < Ni(e/2) (see also Exercise 12.2). Therefore, for any
e>0,
£log(l+logiVi(2e)) <K^T),
i = 1,..., m, and so for any e > 0,
m
N(e) < J2 W(0 < m (exP (e2iC'~ V* - l)) .
Therefore
elog(l + logiV(e)) ^elogflogm + e2*'"'"/^

550 BOUNDEDNESS AND CONTINUITY VIA METRIC ENTROPY 12.4
and
iim£_0 elog (1 + log JV(e)) < 2K^lrj.
Letting 7? -+ 0 yields (12.3.6).
If 1 < a < 2, the same argument (using (12.3.1)) proves (12.3.5). This
completes the proof of the theorem. I
Although the "gap" between the result of Theorems 12.2.1 and 12.3.1 is wide,
we saw that the sufficient conditions of Theorem 12.2.1 cannot, in general, be
improved. Similarly, one cannot hope to improve significantly the necessary
condition of Theorem 12.3.1. For example, if {X(t), 0 < t < 1} is a real
stationary harmonizable SaS process (see Section 6.5), then it is sample bounded
and sample continuous if and only if
[ (\ogN{e))^a'de<oo, 1< a < 2, (12.3.10)
Jo
and
/ log(l+log7V(e))de<oo, a = 1, (12.3.11)
Jo
(see Marcus and Pisier (1984) and Talagrand (1990)). One has in this case a
perfect extension of the necessary and sufficient condition (12.1.2) for sample
boundedness and sample continuity of stationary Gaussian processes. The
relations (12.3.10) and (12.3.11) are in fact necessary for the sample boundedness and
continuity of any stationary SaS process, 1 < a < 2 (Nolan 1991).
Finally, we note that even majorizing measures, a more complicated tool than
metric entropy, fail to produce necessary and sufficient conditions for sample path
continuity and boundedness of SaS processes with 1 < a < 2 (Talagrand 1988).
12.4 Boundedness and continuity of self-similar a-
stable processes
In this section, we collect a number of facts, scattered throughout the book, which
are related to boundedness and continuity of a-stable if-self-similar processes.
By Corollary 7.1.11, the admissible values of (a, H) lie in the domain
P={0<a<2, 0< H <max(l,a-1)},
illustrated in Figure 12.1. Divide this domain into the following regions:
Ai = {(H,a): a = 2, 0<H< 1/2},

12.4
BOUNDEDNESS AND CONTINUITY OF SELF-SIMILAR PROCESSES
551
A2 = {(H,a): 0< a < 2, 0<H< 1/2},
B, = {(H,a): a = 2, 1/2 < tf < 1},
B2 = {(#,<*): l/H<a<2, 1/2<H< 1},
B3 = {(#,a): a= 1/tf, \/2<H< 1},
B4 = {(#>): 0<a<l/i7, 1/2 < if < 1},
C, = {(H,o): a=l/H,H>l},
C2 = {(J?,a): 0<a< 1/ff, H> 1}.
1 2
Figure 12.1: The domain V.
The corresponding process can be either:
• sample continuous,
• discontinuous with sample paths which are bounded on any compact interval,
• unbounded on any interval of positive length.
We shall now examine what happens in the various regions of V.
Surprisingly, when 0 < a < 2, the region B2 is the only one where one can
make a general statement on sample continuity:
Theorem 12.4.1 Let {X(t), t > 0} be a strictly a-stable, 1 < a < 2, H-sssi
process such that
-<H<1, (12.4.1)

552 BOUNDEDNESS AND CONTINUITY VIA METRIC ENTROPY 12.4
that is, suppose (a, H) £ Bj- Then {X(t), t > 0} is sample continuous.
This theorem is a generalization of Example 12.2.3, and its proof is identical to that
of the example: In regions A\ and B\, the process is fractional Brownian motion,
which is, as we know, sample continuous (Exercise 10.1). If A is a SQ/2(1,1,0)
random variable independent of the H-sssi fractional Brownian motion Z, then
the subordinated process X(t) = A^2Z(t), t > 0, is clearly SaS, #-sssi, and
sample continuous. This gives examples of sample continuous processes in the
regions A\, B\, A%, B2,2?3 and B4 of V.
The well-balanced linear fractional stable motions with H < \/a, and the
Takenaka process are examples of processes in regions A2 and £?4 which are
not sample bounded on any interval of positive length. The log-fractional stable
motion is an example of a process in B-$ whose sample paths are unbounded on
any interval of positive length. (See Examples 10.2.5, 10.2.7 and 10.2.6).
Of course, the only SaS H-sssi process in region C\ is the SaS Levy motion
which is not sample continuous, but is sample bounded on compact intervals (see
Exercise 9.5).
The following two examples show that region C2 contains both sample
continuous processes and processes which are unbounded on intervals of positive
length.
Example 12.4.2 We want to construct a SaS ff-sssi sample continuous process
{X(t), t > 0} with (a, H) in region C2. Our construction is based on a particular
family of iJ-sssi processes due to Kesten and Spitzer (1979). Let U\, U2,. ■.
be a sequence of (say) symmetric i.i.d. integer-valued random variables in the
domain of normal attraction of a symmetric 0-stable law, 1 < 6 < 2, and let
£(1), £(2),... be a sequence ofi.i.d. real random variables in the domain of normal
attraction of a /?-stable law, 0 < (3 < 2. The two sequences are independent. Set
Sk = U\ H h Uk, and let Wn = Yll=\ £(5k)- T^11 tne sequence of stochastic
processes {Dn(t) — n~sWnt, t € [0,oo)} (with Ws denned, for a non-integer
s, by linear interpolation) converges, as n —> 00 weakly in C[0, 00), to a sample
continuous 6-self-similar process {A(t), t > 0} with stationary increments where
6=1-1/6+ 1/(0(3) (Kesten and Spitzer (1979)).
The law of A depends only on 8, (3 and the tails of f/iS and £(j)s, i.e., on
lirn^oo \eP(Ui > A) and lim*-^ A"P(±£(j) > A). Further, it is easy to
show that for every t > 0 and every 0 < p < (3,
£|A(i)|p < 00.
We now construct a SaS if-sssi process with a given (a, H) in region C2 as
follows. In that region, 0 < a < 1 and 1 < H < I/a. Choose a/5e (a, 1) such

12.4 BOUNDEDNESS AND CONTINUITY OF SELF-SIMILAR PROCESSES 553
that
„ 1-/3 1 r x
Let { A(i), t > 0} be the corresponding Kesten-Spitzer process with all continuous
sample paths, defined, say, on a probability space (£2i,.Fi,Pi). Letting M be a
SaS random measure on (Cl\, T\) with control measure Pu we now define
A"(i) = / A(t,W|)M(dwi), i>0. (12.4.2)
Then {X(t), i > 0} is a 5q5 if-sssi process (Theorem 7.9.1). Since (A(i), £ >
0} has continuous sample paths and 0 < a < 1, sample continuity of {X(t), t >
0} will follow from Theorem 10.4.2 once we show that
E\ sup |A(i)|Q < oo. (12.4.3)
0<t<l
In terms of the approximating processes Dn, it is enough, of course, to assume
that the £(x)s are themselves S(3S, and then to show that for every 0 < p < ft,
there is a constant C < oo such that, for every n > 1,
E sup \Dn(t)\p < C. (12.4.4)
0<«<1
Defining Ni (x) as the number of times the random walk { Sk, k > 1} visits x up to
time i, we denote for each n > 1 and isZ.a random vector N(n'x) G C[0,1] by
N("'x)(i/n) = Ni(x) for 2 = 0,1,..., n, and with linear interpolation between
these points. Then
E sup \Dn(t)\* = n-i>sE sup I T S(x)N[nt](x) *
OO
< n-pSE\\ ]T ^(a:)N(n'x)|P. (12.4.5)
x= —oo
Write the expectation in the right-hand side of (12.4.5) as E^E(. For fixed
^N(n,x)j x € z}, the vector Y = Exl-oo £(z)N(n-x) is a Sf3S random element
of C[0,1] with spectral measure
oo .
rY = 22 ||N X)|| 2 (*{N(».*>/||N<».=)t|} +(5{-N(».=)/||N(".-)||})
x~—oo
OO j
= 2_/ ■^n(:C) 2 (^{N<"-*)/||N(".»)||} + *{-N<».»>/||N(".«>||}) •

554 BOUNDEDNESS AND CONTINUITY VIA METRIC ENTROPY 12.4
Thus for any <j> in the dual space,
• £e^Y>=exp{- J |0(x)prY(dx)},
where B\ is the unit ball in C[0,1].
Now C[0,1] is a Banach space and all Banach spaces are of stable type
P e (0, l).1 Applying the inequality (12.4.6) with E = C[0,1], ||-|| = supremum
norm and 0 < P < 1, we conclude that given 0 < p < /? < 1, there is a constant
C\ < oo such that
oo
E,\\Y\\P ^ E\\ J2 €(*)N^)f
x= —oo
< c( £ iiN<"...ir)"s
x=—oo
I=— OO
Hence, for every n > 1,
£7 sup |Dn(t)|p < Cin-rsE( V Nn(x)PY • (12.4.7)
o<t<i V *-*' J
Let f£T„ = maxi<n \Si\. Then JV„(a:) = 0 for every |x| > Kn. By Holder's
inequality (0 < P < 1),
( £ ^(x)'3) < (2irB+l) ^ iVn(x) = \2Kn+\) n.
x= —oo a:= — Kn
(12.4.8)
It is now sufficient to show that for every 0 < r < 6 there is a constant C^ < oo
such that for every n > 1,
EKrn < C^nr>e (12.4.9)
1 Let fi be i.i.d. SfiS random variables with 0 < 0 < 2. Then the sum J^ _0 £iXi converges a.s.
if and only if T^^ln W < °°- This property does not extend to arbitrary Banach spaces. Let E
be a Banach space with norm |j • ||. The Banach space E is said to have stable-type p if J^._ &X{
converges a.s. for all X;s in E such that ^2°1, W'X-iW0 < oo. No E i=- {0} has stable type greater
than 2. A Hilbert space has stable type 2. All Banach spaces have stable type /3 £ (0,1).
The Banach space E has stable type /9 if and only if for some (each) p 6 (0,/?),
(4t^W)'" HtlNf)'" <«*>
forsomeO<c<oo independent of n and all X i,..., Xn 6 E. For further detai Is see Linde (1986).
For proofs see Schwartz (1981).

12.4 BOUNDEDNESS AND CONTINUITY OF SELF-SIMILAR PROCESSES 555
because then (12.4.3) follows from (12.4.7), (12.4.8) and (12.4.9) withr = p(-l +
To prove (12.4.9), note that by Levy's inequality,
EK^ = Ema.x\Ul + --- + Ui\r
i<n
< 2E\Ui + ■ ■ ■ + Un\T = lEtEu^iUx + ■■■ + enUn\r,
where the 6jS are i.i.d., independent of the UiS, and Rademacher, that is, P{t\ =
1) = P{e\ = -1) = 1/2. By Khinchine's inequality (Exercise 11.1),
lEMnUx + ■■■ + enUn\r < C2E{U2 + ■■■ + U2n)rl2
for some C2 < co. Since P{Ui > A) = 0{\~8) as A -► oo, P(U2 > A) =
0{\-6'2) and therefore
^<c3(i + y,)
(stochastic ordering), where V\ ~ Se/2{\, 1,0). Using the stability property of
the Vs,
E(u2 + --- + uly'2 < c;/2E(n + vl + v2 + --- + vny'2
< Cll2E(n + n2leV{)r>2
< C4nr'2 + nr'eEV[/2
< C,nT'6
for some C5 < 00. This concludes the argument.
Example 12.4.3 We shall construct a SaS ff-sssi process {X(t), t > 0} with
(a, H) in region C2 which is not sample bounded on intervals of positive length.
Given an (a, H) in region C2, set
a, = (|y/2e(ay/2]c(0,l),
Hi = (aH)x'2 e [ax>2,1) C (0,1).
Let {Y (t), t > 0} be a symmetric linear fractional 1-stable motion with index of
self-similarity Hi. Note that this process is not sample bounded on every interval
of positive length. Now let {Z(t), t € 1R} be an increasing strictly Qi-stable
Levy motion, independent of {Y(t), t > 0}. The two processes are assumed to
be defined on probability spaces (Qj,T\,P\) and (Ql2, F2, P2), respectively, and
we choose separable and measurable versions of these two processes.
This is always possible by Theorems 9.2.5 and 11.1.1. Define on the product
probability space the composition process
W(t) = Z(Y{t)), t > 0.

556 BOUNDEDNESS AND CONTINUITY VIA METRIC ENTROPY 12.5
It follows by Exercise 7.1 that {W(t), t>0}isHi- (1/q,) = tf-sssi. Further,
for every 0<a<6<oowe have supa<t<5 \Y(t)\ = oo a.s. Therefore, for
P\-almost every wj there is, say, a countable set tut2>... in (a,b) such that
l^(OI —+ oo as n —+ oo. Thus,
P2( sup \Z(Y(t))\ = oo)>P2( sup \Z(s)\ = oo) = 1.
Ka<t<b ' Kse{Y{tn), n>l) '
Therefore
PlxP2( sup \W(t)\ = oo) = 1. (12.4.10)
Ka<t<b '
Note also that since a < a>\, we have
EixE2\W{t)\a<oo (12.4.11)
for every t > 0.
Now again let M be a SaS random measure on (Q, x Q2, T\ x T2) with
control measure P\ x P2. Relation (12.4.11) ensures that the process
X(t)= f I^(t,(w1,W2))M(d(a;1,w2)), t > 0,
is well defined. Moreover, {X(t), t > 0} is a SaS if-sssi process, and since by
(12.4.10), E, x Eh supa<t<6 | W(i)|Q = oo for every 0 < a < b < oo, it follows
from Theorem 10.2.3 that {X(t), t > 0} is not sample bounded on intervals of
positive length.
12.5 Exercises
Exercise 12.1 Let h: (0, oo) —* (0, oo) be a non-increasing function such that
h{6) T oo as 6 i 0. Show that there is a SaS process {X(t), 0 < t < 1}, 1 <
a < 2, such that {X{t), 0 < t < 1} is not sample bounded, and for every
O<0< 1,
/ N{eY'ade < h(9),
Je
where N(e) is the covering number associated with the metric dx generated by
the process.
Hint: Go through the following steps:
(i) Show that it is enough to consider only piecewise constant functions h
taking values in the set {1,2,3,...}. Namely, there is a sequence 1 = C\ > C2 >
• • • —► 0 such that
h{d) = k if Ck+] <6<Ck, k= 1,2,....

12.5
EXERCISES
557
Moreover, the sequence {Ck, k = 1,2,...} may be chosen in such a way that
C^a is a positive integer for any k = 1,2,....
(ii) Let X(0) = 0, and for ^ < t < I, X(t) = anXn, n = 1,2,...,
where X\,Xi,.. ■ arei.i.d. 5a(l,0,0) random variables, and {an, n= 1,2,...}
is a sequence of positive numbers with ai = 1, decreasing to 0. Show that
{X(t), 0 < t < 1} is sample bounded if and only if J^Li an < °°-
(iii) Show that for any an+\ < 6 < an, N(9) =n+ 1, and
/ N{ey'a<k<Yi(ak-ak+l)(k+iy'a, n=l,2,....
Je it=i
Let bn — h(an), n = 1,2,... . Conclude that it is enough to find a sequence
{an, n = 1,2,...} such that
n
J2(a*-ak+0(k + l)1/a<bn, n=l,2,... ,
fc=i
and E~=i < = °°-
(iv) For a function h as in (i), let
an = Ck if mk < n < rrik+i, fc = 1,2,...,
where 1 = m,\ < mi < 7713 < ... is a sequence of positive integers that tends to
infinity. Then bn = k if ro^ < n < km+\. Conclude that it is enough to choose
the sequence {m.k, k= 1,2,...} in such a way that
k
J2 (Ci - Ci+^mli* < k, fc=l,2,... ,
i=l
and
00
Y^C^{rnk+i -mk) = 00.
fc=i
(v) Show that the choice mi = 1,
fc-i
mk = Y2C~a, i = 2,3,... ,
i=i
works.
Exercise 12.2 Let C be a set, and d a (pseudo) metric on C. For an e > 0 define
N(C,e) = the minimal number of open d-balls of radius

558 BOUNDEDNESS AND CONTINUITY VIA METRIC ENTROPY 12.5
e centered in C, covering the whole of C,
M(C, 6) = the largest number of points in C such
that the distance between any two different
points is at least e.
Let C\ C C. Show that
N(Ci,e) < M(Ci,e) < Af(C,e) < N(C,e/2).
Exercise 12.3 Let X\,X2,■ ■ ■ be non-negative random variables such that for
some/3 > 1,
P{Xi>x)<e-x\ x>0, t= 1,2,....
Show that there is a finite number Kp such that for any n = 1,2,...,
£max{X,,... ,Xn} </^(l + logn)'/'3.

Chapter 13
Integral representation
Let {X(t), t e T} be an a-stable process. We have seen in the previous chapters
that it is very convenient to study the properties of such a process via an integral
representation of the type
X{t) = / f{t,x)M(dx) + rj(t), t 6 T.
JE
Here, our goal is to determine the a-stable processes that can be so represented.
We shall also examine whether convenient choices of E and M are possible, for
example, E = (0,1) and M having Lebesgue control measure. We may also want
M to be symmetric if X is SaS and to have skewness intensity /3(-) = 1 if X is
skewed a-stable.
Because the construction of the integral representation involves many steps,
we start by describing the main ideas. Suppose firstly that T is countable. The
spectral representation of an a-stable random vector (X\,X2,- ■■ ,Xd) in Md
involves a spectral measure T on Ed. Since we want to treat all ds simultaneously,
it is best to establish a representation for a vector X = (X\,X2, ■ ■■) with an
infinite number of components. To accomplish this, we must view X as a vector
in a suitable Banach space. We shall suppose limi_oo Xi — 0 a.s., so that X € co,
the Banach space of all sequences of real numbers converging to 0, with norm
||x|| = sup^ \xi\. Proposition 13.1.1 below states that there is a measure Y on the
unit ball B\ of cq that can be used to represent any finite-dimensional distribution
ofX.
The resulting representation is defined on E = B\, the unit sphere of Co, and
not on the interval E = (0,1). To pass from B\ to (0,1), we make a change of
variables. This has to be done carefully because the measure T may have atoms.
Write T = Ta + rc where Ta is the atomic part of the measure. ra is defined
on a subset A of B\ and Tc on Bi\A Firstly, map the set B\\A where rc is

560 INTEGRAL REPRESENTATION 13.1
defined, into the interval (0, j]. The map is chosen so as to preserve measurability
and to transform rc on B\\A into Lebesgue measure on (0, 5]. (Theorem 14.3.9
of Royden (1963) is used for this purpose.) Secondly, spread the discrete mass
of A uniformly on (j, 1). This gives the desired representation, but only for X
satisfying limt_oo Xi = 0 a.s.. To consider a general a-stable X = (X\, X2,...),
set Yi — a,iXi with aj J. 0 fast enough to ensure that Yi —+ 0 a.s. as i '—» 00.
Apply the previous argument to Yi to obtain an integral representation involving an
integrand, say &, then set fi = a~xgi to obtain the desired integral representation
for Xi. The result is formally stated in Theorem 13.1.2 below.
We must finally consider an arbitrary index set T, not merely a countable
one. If we insist on E = (0,1) and Lebesgue control measure, then we must
suppose that {X(t), t € T} satisfies Condition S. This condition, introduced in
Section 3.11, states that X(t) is separable in probability, i.e., there is a countable
set To = {ti,t2,...} C T so that at each t € T, X(t) can be approximated in
probability by X(tik) where **,, t»2, - - - € To. Since the integral representation
holds for {X(tik)}, it will hold for X(t). The result is stated in Theorem 13.2.1
below for any a if {X(t), t € T} is SaS but only for a ^ 1 if {X(t), i € T} is
skewed. It is not known whether it holds in the skewed a = 1 case.
What happens if {X(t), t € T} does not satisfy Condition S? Then the choice
E — (0,1) and Lebesgue control measure is impossible, but Theorem 13.2.2
states without proof that in the SaS case, an integral representation exists, for
some space E and some control measure m.
Let us now fill in the details.
13.1 Countable parameter space
We suppose in this section that T is countable. The following proposition assumes,
in addition, lim^oo Xj = 0 a.s.
Proposition 13.1.1 Let {Xi, i = 1,2,...} be an a-stable process satisfying
lim Xi = 0 a.s..
i—*oo
Let Co be the Banach space of all sequences of real numbers converging to 0.
Then there is a finite measure T on the unit sphere B\ of cq and a constant
vector fi = (/X|, /i2, • • ■) € Co such that, for any n = 1,2,... and real numbers
6\,...,9n,
n
Etxp[iY^ejXj} =

13.1
COUNTABLE PARAMETER SPACE
561
C n a n n
exp{-/ \l29ixi\ (1-isi«n(X]^)L(a!:£^))r(dx)
n
i=i
where, as usual
(tan(7ra/2) if a ^ 1,
(13.1.2)
-(2/7r)ln|a| ifa = l.
PROOF: By assumption, X = {X\, X2, ■ • •) is a random vector in Co- We may
and shall assume that it is not an a.s. zero vector. Let Px denote its distribution.
For a given 0 < r < a, let P£' be a finite Borel measure on c$, absolutely
continuous with respect to Px, defined by
dP!
(r)
X
dPx
(x)
The measure P-£' is finite by Proposition 10.2.1. Now define a finite Borel
measure Tr on the unit sphere Si by
rr = (a-r)4r)or11 (13.1.3)
where T: co\{0} —► B\, defined by Tx = x/||x||, maps x into the unit ball B\.
Note that the total mass |r,.| of TT equals
|rr| = (a -r) f ||x||rPx(dx) = (a - r)E sup \Xi\T.
Applying Proposition 10.2.1 together with the argument leading to Property
1.2.18 gives 0 < inf0<T-<a |IV| < sup0<r<Q |rr| < oo. Choose now a sequence
{rn, n > 1} of positive numbers increasing to a such that |rrJ converges to a
finite limit as rn f a, say, |ra|. Do the measures TTn converge as well? Clearly,
{ir*-rrr„, n > 1} is a family of probability measures on co supported by the
unit sphere B\. We claim that the sequence mn = p^-|rrn, n > 1, is relatively
compact in the topology of weak convergence, i.e., every subsequence of mn has
in turn a convergent subsequence. By Prohorov's theorem, it is enough to show
that the sequence mn is tight. Regard {0,1,2,..., oo} as a compact metric space
with a single cluster point. By the Arzela-Ascoli theorem, it is enough to show
that for each positive numbers e and 77 there are integers J and no such that, for
each n > no,
mn{x 6 Co: sup|xi| > e} < n. (13.1.4)
i>I

562' INTEGRAL REPRESENTATION 13.1
To prove (13.1.4), note that
mn(x E cq: sup|xi| > e)
= 1Ft/ W"Px(dx)
|lr„| ^{xtsup^Jx.l^HxIl}
<const(o-r„)Bf||X||r»l(sup|Xi|>e||X||))
v «>/ . '
<conste"-r'>(a-7"n).Esup|Xi|r*\ (13.1.5)
The argument used in Property 1.2.18, when applied to Corollary 10.2.2, gives
lim limn_00(a - rn)Esup \Xi\Tn = 0.
/—»oo
i>I
Relation (13.1.4) now follows from (13.1.5), and so the sequence {mn =
J-rrrti} is, indeed, tight. There is therefore a probability measure Q on c$
such that for some subsequence {r„fe, k = 1,2,...}, |rrn |-1rr„ =4- Q. Since
the measures rrn are concentrated on the unit sphere B\, so is Q. Define
r=(aCa)-1|ra|Q. (13.1.6)
To obtain fx express the distribution of each Xj as Xj ~ Sa(&j, Pj, Vj)> and set
,, - / Vj ifa^ lf m i 7^
^'{Vj + Ub^J^B \sj\r(ds) ifa=l, [iiAJ)
for j = 1,2,... . We claim that T and /j constructed in (13.1.6) and (13.1.7)
satisfy (13.1.1).
Firstly, we need some notation. Fix d G {1,2,...}. Let sj'"°° =
{x: maxj=i d \xi\ — 1} be the unit sphere relative to the L°° norm on Rd and let
T^ and /j,°'(d) be the parameters in the representation of the characteristic function
of the a-stable random vector (X\,..., Xd), given in Proposition 2.3.8. (To be
consistent with the notation of Proposition 2.3.8, we should write rf, 1 and Mii'.Ji^
instead of T^ and /jP'(d\ but this is too cumbersome.) It is important to
distinguish below between the norm ||-|| on cq (defined as ||x|| = supi=1 2 \xi\, x£co)
and the norm ||-HooOnR^ (defined as Hxlloo = maxi=h_td \xi\, x S Md), between
the unit spheres B, = {x € cq: ||x|| = 1} and sj'"00 = {x G Rd : ||x||oo = 1}
and between the measures T on B\ and r^d) on sj'"°°. The usual spectral measure
Td on the unit sphere Sd (defined with respect to the Euclidean norm) also appears
below.

13.1
COUNTABLE PARAMETER SPACE
563
Suppose firstly a ^ 1. By virtue of Proposition 2.3.8, it is sufficient to prove
f |X>*,rr(dx)= f \J2e^\ar(d)(ds) (13.1.8)
■J D\ ■ i J S , i
'* i=i Jsy- i=i
and
/ lE^I<Q>r(^)= / |^^Sj|<Q>rW(ds). (13.1.9)
Since the integrand in the left-hand side of (13.1.8) can be extended to a bounded
continuous function on Co,
/ I^T^rndx) = (aCa)-1 lim [ iTOiX^Trldx)
= (aCa)-' lim / |r^,-hrr (dx.)
r d
= {aCayx lim (a - r„J / | J] fl^f «* Px(dx)
d
= {aCa)~l lim (a - rnk) E^OjX^
j'=i
a" „ (Property 1.2.18)
l.jt9^^
>sd j=\
proving (13.1.8). Since (13.1.9) follows in the same way, this completes the proof
of (13.1.1) in the case q^I.
Consider, now, the case a = 1. With the same notation as before, (13.1.1)
will follow from (13.1.8) and
« /• d d d
— / (E 9w) lo§ i E ew ir(dx)+E 6m
(13.1.10)
Since the preceding prooffor(13.1.8) worksfora = 1 as well, only (13.1.10) must
be verified. We use Corollary 4.4.9 and the fact that Y?j=\ 6jSj log | ]T^=1 OjSj\

564 INTEGRAL REPRESENTATION 13.1
is a continuous function on S^'00. We have
d d
JB< j=\ j=l
. d d
= ?tlim / (X^'x0logiX^'xjir'-"*(dx)
/* ^ <r > ^
= fj!™,/ QC^i) "* lo8l53^'lrr-*^ (sincer„fc |1)
/B' j=i j=i
d
= tlim aw / (E^^)^"^los| X>i§r p*^
(set ar = (7r/2)(l - r) and use (13.1.3))
d
+&^ /.(g^P'M =^)*c*o
- &^I(£»'EfcP>H£»<i£
j = l """~ j=l
\\x\\™kP(.x xd)(dx)
(here || • H^ is the supremum norm in Rd)
+,5- -,I(t>>%)^(,3Sh)Mr-*<*■>
(since rnfc T 1)
d d
•^ 3 = 1 j=l
+ (y9jXj)log( max |x,|)r(dx) (use (13.1.3)).
From Proposition 2.3.8 and Example 2.3.4, verification of (13.1.10) reduces to

13.1
COUNTABLE PARAMETER SPACE
565
showing that for any j=l,2,...,d,
f fma-x-i=i,...,d\xi\\ W j \ f i /maxi=i...,d |sj|\
/ x; lQg nrf r(dx) = / sj l°s , I Td(ds),
Jb, \ \xj\ / Jsd \ \Sj\ J
(13.1.11)
where Tj. is the spectral measure of the vector X\,..., Xd (i.e., with respect to the
Euclidean norm). However, (13.1.11) follows as above by a simple application
of Corollary 4.4.9. Thus, we have established (13.1.1) in all cases.
It remains only to show that fi. € c0. The case a ^ 1 being trivial, only the
case q = 1 must be considered. Applying Theorem 10.3.1, we conclude that
lim [fij / i,-ln|a;,-|r(dx)) = 0,
j—oo\ 7T JBx J
and now the bounded convergence theorem applies and gives limj_»oo fij = 0.
This completes the proof of the proposition. I
We are now in position to prove the representation theorem when T is
countable.
Theorem 13.1.2 Let T = {1,2,...} and {Xi, i S T} be an a-stable process.
(i) There are functions fi € F, i — 1,2,..., and real numbers fa, i =
1,2,..., such that
{Xui=\,2,...} = U fi(x)M{dx) + fM, 1=1,2,...}, (13.1.12)
L°(0,1) ifa^l,
{/:/01|/(x)ln|/(x)||dx<ac} ifa=l,
and M is an a-stable random measure on ((0, \),B) with skewness intensity
f3 = 1 and Lebesgue control measure.
The functions fi in (13.1.12) can be chosen uniformly bounded if
limj-.oo Xi = 0 a.s.
(ii) If{Xit i = 1,2,...} is strictly a-stable and a ^ 1 then the representation
(13.1.12) holds with fa = 0.
(Hi) lf{Xu i = 1,2,...} is SaS, then
{Xi, i= 1,2,...} = [J fi(x)Ms(dx),i=l,2,...},
where fi e La{0,1), i = 1,2,..., and Ms is a SaS random measure on
((0, l),B) with Lebesgue control measure. The functions fi can be chosen
uniformly bounded //limj—oo Xi =0 a.s.

.*»<>(>
INTEGRAL REPRESENTATION
13.1
Proof: For i = 1,2,... , choose a. > 0 such that P{a~x\Xi\ > i~l) < 2_i.
Let Yi = a~'Xj, i — 1,2,... . Then {Yi, i = 1,2,...} is an a-stable process
satisfying lim;_oo Yi — 0 a.s.. We apply now Proposition 13.1.1 to conclude that
there is a finite Borel measure V on the unit sphere B\ of cq and a vector 77 6 co
such that for any n = 1,2,... and real numbers 6\,...,6n,
n
n n n n
£ejXjr (1 - i sign (]Teixl)L(Q; E9>xi))r(dx) - * E*i»fe
with L given by (13.1.2).
We now decompose T into "atomic" and "continuous" components. Let
A — {x: T({x}) > 0}. Since T is a finite measure, the set A is at most
countable; we will denote its points x^, i — 1,2,.... Write r = Ta + Tc where
Ta{C) = T{C n A), rc(C) = r(C n Ac) for any Borel set C C Bx. Since
rc is an atomless finite Borel measure on B\, Theorem 15.9 of Royden (1963)
applies, and we conclude that there is a Borel set Co in B\ with Tc{Cq) = 0, a
Borel set Jo Q (0> 5] of Lebesgue measure 0 and a one-to-one map <p from B\ \Co
onto (0, \)\h with both <fi and cf>~x Borel measurable, that preserves the measure
(2rc(51\C0))_1 Tc. That is, for any Borel set D C (0, \\\I0, we have
\{D)=Tc{r\D))/2Tc{Bx),
where A denotes the Lebesgue measure on (0, j]. Foreachj > 1, define a function
5f:(0,i]-,Eby
f (0-,(y)),(2rc(JB1))'/- ifyeCO.UVo,
4')(2/) =
{ 0 ify€/o.
Clearly, 5} \g^ ,... are measurable uniformly bounded functions.
Construct also the intervals:
i = 1,2,..., and use them to define a sequence of piece wise constant functions
gf*: (i,l) -» R as follows:
fo ifye(i,i)\(U>,^),
sf }(y) = {
i (2rQ(B,)),/Q4l) ifi/e Ji, t = i,2,...,

13.1
COUNTABLE PARAMETER SPACE
567
where x(i) = (x[^, x^,...), i = 1,2,..., are the points of the set A defined
above. g\ , g\ ',... are measurable uniformly bounded functions because x^' €
Bt. Now combine the two sequences into a single sequence of measurable
bounded functions on (0,1),
9j(y) = <
g^(y) ify 6(0,1],
(13.1.13)
^ gf\y) ify€(i,l),
j'■ = 1,2, Let M be an a-stable random measure on ((0,1), B) with skewness
intensity /? = 1 and Lebesgue control measure. We claim that
{Yj,j^l,2,...,}£{JXgj{y)M(dy)+Jlj,j=l,2,...}, (13.1.14)
where
Vj,
tj=={ r 1
77,- - I iog(2re(B,)) JBi Xjrc(cbc) + iog(2ra(B,)) /B| x,-rB(dx)j,
(13.1.15)
for a 7^ 1 and a = 1, respectively. Indeed, for every n = 1,2,... and any real
-ln£exp{z^^ / flj-(i/)M(dy)j
= / |X>^(y) °(l -isign ^9igi{y))L{a-,Ydeigi{y)))dV
/•1/2 /•!
= / +/
./0 J1/2
i^^|f:^(2rc(JB1))1/a^|Q
(l-isign (^^(2rc(S1))1/QxJ)L(a;^^(2rc(JB1))1/a^))rC(d:
j=i j=i
-i— £r({x«})[ f)eJ-(2r0(BI))'/-«5*)|a
(l -isign (X:^(2ra(Sl))'/Qxf )L(a;f:^(2ra(fl,)),^fc)))
TTr.
X 1 -
+ ;
(13.1.16)
be)

568 INTEGRAL REPRESENTATION 13.2
Ifa^ 1, (13.1.16) implies
- In £exp{i J2 ®i (J 9j(y)M(dy) + £,) }
. n n
= / E^'l (i-*sisnE^a;j)tan^)rc(d,c)
JB< J = l j=l
C n a n n
+ 1 \J20ixj\ (l -{sisn (520ixj)tanY)Fa^dx)~{Y,ew3
1 J = l 3 = 1 3 = i
- n n n
= / \%l0ixj\ (1-i*wfe6jXJ)tm^y(dx)-i'jr/0jrij
JB> i=i i=i z j=i
n
= -ln2Sexpji^0,-y;-},
J=l
proving (13.1.14) for a ^ 1. The case a = 1 is similar. Of course, (13.1.14)
implies (13.1.12) with /,- = o^-, /it, = a,^, j = 1,2, — (If limj—oo Xj = 0
a.s., then the choice a,j = 1 is possible and the functions /_,• — gj are uniformly
bounded.) This proves part (i) of the theorem.
The claim of part (ii) follows immediately from part (i) because of Proposition
3.4.3 and Theorem 2.4.1. The proof of part (iii) of is similar and simpler than that
of part (i). I
Remarks.
1. The representation theorem in Rd (Theorem 3.5.6) was stated without proof.
It is a consequence of Theorem 13.1.2 because a vector (X{,..., Xd.) in Rd
can be viewed as (Xt ,...,Xd,0,0,...).
2. A SaS vector admits a representation with respect to a totally skewed
random measure M as well as a representation with respect to a SaS
random measure Ms-
13.2 Arbitrary parameter space
We now consider an arbitrary index set T and determine how widely applicable
is an integral representation with E = (0,1) and M having Lebesgue control
measure. We suppose 0 < a < 2 in the SaS case, but a ^ 1 in the skewed case.

13.2
ARBITRARY PARAMETER SPACE
569
Theorem 13.2.1 Let {X(t), t € T} be an a-stable stochastic process satisfying
Condition S (see Definition 3.11.2).
(i) Let a ^ 1. Then there are functions f(t, •) € La(0, l),t€ T, and real
numbers p,(t), t eT, such that
{X(t),teT}±{J f(t,x)M(dx) + n(t),teT], (13.2.1)
where M is an a-stable random measure on ((0,1),S) with skewness intensity
P = 1 and Lebesgue control measure. Moreover, every a-stable process (even
with a = 1) with an integral representation (13.2.1) must satisfy Condition S.
(ii) Let a y^ 1. lf{X(t), t € T} is strictly a-stable, then the representation
(13.2.1) holds with p,(t) = 0.
(Hi) If {X{t), t € T} is SaS, then
{X(t), t&T}±{f f(t,x)Ms(dx), t € t}, (13.2.2)
where f(t, •) 6 La(0,1), t e T,andMs is a SaS random measure on ((0, l),B)
with Lebesgue control measure. Moreover, every SaS process with an integral
representation (13.2.2) must satisfy Condition S.
PROOF: Let To = {t\, ti,...} be a countable subset of T from the definition of
Condition S.
(i) By Theorem 13.1.2 there are functions fti e La{0,1), i = 1,2,..., and
real numbers p,t{, i — 1,2,..., such that
{X(*i), *=1,2 >={/ fti{x)M(dx) + nu, i= 1,2,...}, (13.2.3)
where M is an a-stable random measure on ((0, l),B) with skewness intensity
P = 1 and Lebesgue control measure.
We extend the integral representation (13.2.3) to the whole of T as follows.
Fix t e T. Because {X(t), t S T} satisfies Condition S, there is a sequence
frtjjfcli in To such that X(tlk) -> X(t) in probability as k -> oo. It follows
from Proposition 3.5.1 that the sequence {f(tik, -)}fcLi is Cauchy in LQ(0,1); we
let f(t, •) denote its limit in La(0,1). The sequence {M*u)}£Li is a convergent
sequence of real numbers; we let p,(t) be its limit. The family {/(£, •)> M*)}> t €
T, obviously satisfies (13.2.1).
Forthe converse, let {X(t)r t € T} have the representation (13.2.1). Since the
space La{0,1) is separable, the set {/{i, •) € T} C La(0,1) is separable as well.
There is therefore a countable subset T0 C T such that {f(t, •), t € T0} is dense
in {/(£, •), i 6 T}. Proposition 3.5.1 now implies that the stochastic process

570 INTEGRAL REPRESENTATION 13.2
Y\(t) = J0 f(t,x)M(dx), t € T, satisfies Condition S. Since the (degenerate)
process 5^(i) = (*{t), t E T, also satisfies Condition S (Exercise 3.18), it
follows (Exercise 3.20) that the process Y(t) = Y\{t) + Y2(t), t E T and hence
{X(t), teT} also satisfy Condition S. Since the case a = 1 is similar because
the space L log L (Leb) is also separable, this proves part (i) of the theorem.
Part (ii) now follows from part (i), Proposition 3.4.3 and Theorem 2.4.1.
The proof of part (iii) is identical to that of part (i), but uses part (iii) of
Theorem 13.1.2 instead of part (i) of that theorem. 1
The proof of Theorem 13.2.1 implies that an a-stable process {X(t), t ET}
that does not satisfy Condition S cannot be represented in the form
X(t) = / f{t,x)M{dx)+r]{t), tET, (13.2.4)
Je
with the function space La(E,£,m) (or even its subset {/(£, •), t E T}) being
separable. Therefore, the only hope for obtaining an integral representation for
an a-stable process which does not satisfy Condition S is to use a non-separable
space La(E, £, m). (The separability of the space depends on E and m.) The
following result, due to Bretagnolle, Dacunha-Castelle and Krivine (1966) in the
case 1 < a < 2 and Schreiber (1972) in the case 0 < a < 1, shows that this is
always possible, at least in the SaS case. It is given here without proof.
Theorem 13.2.2 Let {X(t), t ET} be a SaS process, 0 < a < 2. Then there
is a measure space (E, £, to) and a family {J(t, •), t ET} C La(E, £, m) such
that
{X(t), tET}±[J f(t,x)M(dx), tET],
where M is a SaS random measure on (E, £) with control measure m.

Chapter 14
Historical notes and
extensions
14.1 Notes to Chapter 1
Section 1.1. Augustin Cauchy discovered, in the 1850s that the functions fa
satisfying
/oo
ei6xfa(x)dx = e-°aW, a>0 (14.1.1)
-OO
have the convolution property
(Afa(A-)) * (Bfa(B-)) = CUC-)
for some C = C(A, B) and all A, B > 0 (Cauchy 1853). He was, however, able
to show that /Q(x) > 0 for all x only in the cases a = 1 and a = 2.
To prove the non-negativity of fa it is sufficient to show that exp{—\0\a}
is a characteristic function. Seventy years after Cauchy's discovery, George
Poly a (1923) presented his famous sufficient condition for a function to be the
characteristic function of a probability law: {i>(t), teK} is a characteristic
function if^{t) is real, non-negative, ip(0+) = ip(0) = 1, i(>(t) = ijj{—t) and
tp is convex on (0, oo). The non-negativity of the functions fa in (14.1.1) with
0 < a < 1 follows as a simple application.
Paul LeVy (1924) proved that all /as in (14.1.1) with 0 < a < 2 are non-
negative. Here is a simple argument valid for all 0 < a < 2 found in Durrett
(1991), who learned it from Frank Spitzer.
For any 0 and |x| < 1, (1 - xf = £felo(£)(-x)k where
\k) l-2---k

572
HISTORICAL NOTES AND EXTENSIONS
14.1
Applying this expansion to
i){9) = 1 - (1 - cos 9)a'2, 0 < a < 2,
one obtains
W) = f>fe(cos0)fc with ck = (a/k2\ (-l)fc+1 > 0
since a < 2. We conclude that exp{—\9\a} is a characteristic function by arguing
successively that the functions cos 6, (cos 8)k, ip, and
lim [tlj(eV2n-l/Q)}n = e~Wa
n—»oo
are characteristic functions.
That exp{-|#|Q} is not a characteristic function when a > 2 follows from the
following easily verifiable fact:
Iftp(9) is a characteristic function of a random variable X andif
W) - 1 a2
]$—eT- = -T>-°°'
then EX = 0 and EX2 = a2 < oo (Durrett 1991).
In particular, if V>(^) = l+o(02)as# -► 0, then a2 = 0 and therefore tp(9) = 1. If
a > 2, exp{-|#|Q} = 1 + o(92) and hence exp{-|0|a} cannot be a characteristic
function in this case.
The term stable was coined by Paul L6vy. LeVy (1924) gives not only the
characteristic functions of all SaS laws but also those of all strictly stable laws.
The next step, to pass from strictly to non-strictly stable laws, is of course easy if
tt/1. The case a = 1 was settled by Aleksander Yakovlevich Khinchine and
Paul Levy in (1936). It is worth reading in this regard Sections 31, 41 and 42 of
Levy's mathematical biography (Levy 1970).
Many textbooks contain an introduction to stable laws. See, e.g., Breiman
(1968) and Laha and Rohatgi (1979). Lukacs (1970) devotes a number of sections
to the analytic properties of stable distributions. There is, in many of these
publications, great confusion about the sign of ft in (1.1.6) as noted by Hall
(1980).
The classic introductions to stable laws have been for many years Gnedenko
and Kolmogorov (1954) and Feller (1971). An encylopedic treatment of stable
laws on the real line is given by Zolotarev (1983), a work later translated into
English (Zolotarev 1986). Rossberg, Jesiak and Siegel (1985) provide a good
exposition to the subject. Two more books of interest have recently appeared:
Janicki and Weron (1993) and Christoph and Wolf (1993). The second book is

14.1
HISTORICAL NOTES AND EXTENSIONS
573
about convergence theorems with a stable limit law. It focuses on pseudomoments
(moments with respect to F — S where 5 is a stable distribution and F is a
distribution function in the domain of normal attraction of S), Berry-Esseen type
inequalities, asymptotic expansions and local central limit theorems.
Definitions 1.1.5 and 1.1.6 view stable laws as a special subclass of infinitely
divisible laws. The theory of one-dimensional infinitely divisible laws was
developed by L6vy in 1937; see Levy (1954) and Khinchine (1938).
The SaS distribution with a = 3/2 has the name of Holtsmark, a Danish
astronomer who described, in 1939, a three-dimensional version of this distribution
as a model for the gravitational field of stars (Holtsmark 1939).
Section 1.2. Weron (1984) uses the term "completely asymmetric" for a-stable
distributions with /? = ± 1.
Section 1.3. Hardin (1984) noted the extension of Proposition 1.3.1 to the skewed
case and deduces other facts related to skewed stable laws, which were relatively
neglected at the time.
Section 1.4. A series representation of infinitely divisible random variables
without a Gaussian component (of which the series representation of stable random
variables described in the present section is a particular case) was established by
Ferguson and Klass (1972) and developed by LePage (19906). It is also
considered in Vervaat (1979). An extension of these series representations to more
general situations is developed by Rosinski (1990).
Section 1.6. Probability density function tables of the Q-stable distribution can
be found in Mandelbrot and Zamfaller (1959) for j3 = 1, a > 1, and in Holt and
Crow (1973). DuMouchel (1971) and Brothers, DuMouchel and Paulson (1983)
provide upper tail probability tables. The tables of Appendix A are taken from
the last paper.
Analytic properties of stable densities were a major topic of research in the
1930s, 1940s and 1950s. The following curious relation between stable densities
was discovered by Zolotarev (1954): //1/2 < a < 1 and
X ~ Sa(a(a,0i), A,0), Y ~ S1/o(a(l/a,/32),ft,0),
then, for any x > 0,
/x(x)=x-(Q+1>/y(ar°).
Herea(a,0) = (1 +/32 (tan ^f)2)-l^2a), and p{ and fa are related by
( -no. 7ra(7+ 1)\_
0i = -(tan —tan—^ '-) ,

574
HISTORICAL NOTES AND EXTENSIONS
14.1
for some \i\ < 2 — 1/a.
The a-stable distributions are all unimodal. Establishing this fact was an
exciting and difficult problem, whose complete solution escaped probabilists for
a long time, leaving a trail of errors and counterexamples. Wintner (1936) proved
unimodality of symmetric a-stable laws. Ibragimov and Chernin (1959) claimed
that they proved that all stable laws are unimodal, but Kanter (1976) found an
error in their proof. Finally, Yamazato (1978) closed the problem by proving
that all class L infinitely divisible distributions (including stable distributions as a
subclass) are unimodal.
Section 1.7. The rstab program in Chambers, Mallows and Stuck (1976) is written
in Raftor. It was converted to Fortran by John Nolan.
One must, in general, be careful when calling from S or S-PLUS an external
program which uses real variables (i.e., single precision). This is because S does
not ordinarily distinguish between the "single" and "double" precision storage
modes. When an external function is called from S, the arguments to the function
are passed along as well. It seems that the default storage mode for an non-integer
numeric field automatically becomes double precision. Thus, if a number, e.g.,
0.1, is passed to a Fortran routine which is expecting a real (i.e., single precision)
variable, problems will occur. Unfortunately, in many cases these problems are
not immediately evident. The program does not crash; it just returns the wrong
answers. There are two possible solutions. The simplest is to use only double
precision Fortran/C routines, as we do here. Where this is not feasible, the calls
from S must explicitly use variables instead of merely numbers, and the variables
must be explicitly declared as "single precision". Chapter 7.2 of Becker, Chambers
and Wilks (1988) contains some information on calling external routines from S,
and how the variable passing works.
Procedures for estimating the parameter of a stable distribution include:
maximum likelihood (DuMouchel(1973), Chen(1991)); using the sample characteristic
function (Paulson, Holcomb and Leitch (1975), Koutrouvelis (1980, 1981)
Paulson , Delehanty (1984, 1985)); using the order statistics (Hill (1975), Hall and
Welsh (1984), Loretan (1991)); using tabulated quantiles (Fama and Roll (1968),
McCulloch (1986)). For a review, see Mittnik and Rachev (1993).
Section 1.8. Many standard textbooks characterize the domain of attraction
of a stable law (Theorem 1.8.1). A classic reference is again Gnedenko and
Kolmogorov (1954). The situation is more complicated when one considers the
convergence of normalized partial sums of stationary dependent sequences to
stable laws. One class of techniques uses ideas of extreme value limit theory.

14.2 HISTORICAL NOTES AND EXTENSIONS 575
A typical assumption is that the normalized sum of an "independent imitation"
converges to a stable law. An independent imitation is an i.i.d. sequence with the
same marginal distribution as the stationary dependent sequence. For example,
Davis (1983) considers Xj = G(Yj) where Yj is a stationary Gaussian with
mean zero, unit variance and autocovariance rn = EYoYn and where G is such
that the "independent imitation" Xj of Xj belongs to the domain of attraction
of an a-stable law with 0 < a < 1. He shows that if either rnlogn —> 0
or Yl^-x rn < °°> tnen tne normalized partial sums of Xj also converge to
that stable law. For a recent overview, including various "mixing conditions,"
see Jakubowski (1991, 1993). Another set of techniques which works well for
moving averages is reviewed in Avram and Taqqu (1986a). It includes results of
Astrauskas (1983), Maejima (1983a), and Davis and Resnick (1985). See also
Kasahara and Maejima (1986, 1988). Several of these results involve sequences
with long-range dependence, a dependence structure considered in Chapter 7.
Sometimes a point process setting is useful for proving functional limit
theorems (Resnick 1986). But functional limit theorems can be delicate. Avram and
Taqqu (1992), for example, show that if {Xj} is a "two-dependent" sequence
defined by Xj = tj + ej-\, where the tj are i.i.d., SaS, 0 < a < 2, then as
n —» oo, {n-1/Q Yljli Xj, 0 < t < 1} does not converge in D.([0,1]) endowed
with the usual Skorokhod topology J\. Convergence holds if J\ is replaced by
the other Skorokhod topology M\. Although M\ is weaker than J\, it is strong
enough to make the widely-used functionals maxo<«i and mino<«i continuous.
14.2 Notes to Chapter 2
Sections 2.1 and 2.2. Dudley and Kanter in (1974) claimed that if all linear
combinations of the components of a random vector X are stable, then X is itself
stable. Their argument is, essentially, that of the second remark following the proof
of Theorem 2.1.5. However, this argument works only in the case 1 < a < 2, as
was noted by de Acosta and Kuelbs during a student presentation of the Dudley
and Kanter paper in 1981-82 at the University of Wisconsin, Madison.
Dudley remained interested in the problem, and David Marcus (1983), working
under his direction, constructed the counterexample of Section 2.2. Marcus's
proof was subsequently simplified by Samotij and Zak (1989). The argument of
part (i) of Theorem 2.1.5 which is applicable to the range 1 < a < 2 is due to
Samorodnitsky and Taqqu (1991c).
Section 2.3. The characteristic function of a multivariate stable distribution was
derived by Feldheim (1937) and presented in L6vy (1954).

576 HISTORICAL NOTES AND EXTENSIONS 14.2
Section 2.4. The graphs in Figure 2.1 were obtained by using a Fortran program
written by John Nolan. This program calls a Fortran subroutine written by Jarle
Berntsen and Terje O. Espelid which evaluates two-dimensional definite integrals.
Additional graphs can be found in Byczkowski, Nolan and Rajput (1993).
Section 2.5. Because the term "sub-Gaussian" also refers to a class of stochastic
processes with exponential moments, one may sometimes prefer the alternative
term "Gaussian scale mixture" to denote the random vector in (2.5.3). Here, we
use the term "sub-Gaussian" because it is convenient and has been widely used in
the past: cf. Hardin (1982a, 1982fc), Cambanis (1983), Weron (1984), Marques
(1987), and Maejima (1989).
The fact that non-degenerate sub-Gaussian SaS random vectors cannot be
independent was mentioned (in the case a > 1) by Bretagnolle, Dacunha-Castelle
and Krivine (1966) and proved by Hardin (1982&).
Section 2.6. Weron (1984) uses the term "complex SaS random variable" with
two different meanings. One coincides with our definition, whereas the other
refers to an isotropic SaS random variable.
Section 2.7. The term "covariation" is due to Miller (1978), although the quantity
itself has been used even earlier, e.g., by Kanter (1972).
Section 2.8. The covariation norm was introduced and studied by Cambanis and
Miller (1980, 1981).
Section 2.9. The relation between covariation and James orthogonality was noted
in Cambanis, Hardin and Weron (1988). It first appears in Weron (1984). For a
brief review of other notions of orthogonality, see Desbiens (1987).
Section 2.10. The codifference is related to the function (see (4.7.4)) considered
by Astrauskas (1983). Lemma 2.10.8 has an interesting history. It is due to
Schoenberg (1937) and was reproved by Laurent Schwarz in 1945 (see the footnote
in L6vy (1970), p. 134). Our proof follows Cartier (1971).
The following is another measure of dependence considered by Paulauskas
(1976). Let (XUX2) be SaS, 0 < a < 2. The distribution of (XUX2) is
characterized by the spectral measure T on the unit circle 52- Let (U\, U2) be a
random vector on S2 with probability distribution T = r/T(S2). Because of the
symmetry of T, one has EU\ = EU2 — 0. Paulauskas uses
EU1U2 J* sin24>r (d<p)
P~ {EUfEU^y/2 ~ 2(/Jrcos2</>r(d0)/o'rsin2(/.r(d0))1/2
as a measure of dependence for (Xi, X2) where, T is expressed here in polar
coordinates, p has the following properties valid for all 0 < a < 2:

14.3
HISTORICAL NOTES AND EXTENSIONS
577
(i)-l<P<l.
(ii) If X\ and Xi are independent, then p — 0.
(iii) If \p\ = 1 then X\ and Xz are linearly dependent.
Note that p is the correlation of a Gaussian distribution whose T is the T of the
original stable distribution.
14.3 Notes to Chapter 3
Section 3.2. It was long known that using stochastic integrals with respect to an
independently scattered a-stable random measure is a convenient way to describe
a-stable laws; among the first to use these integrals in the symmetric case were
Bretagnolle, Dacunha-Castelle and Krivine (1966) and Schilder (1970). For the
skewed case, see Hardin (1984).
From a practical viewpoint, a stochastic integral is a linear filter which
transforms i.i.d. noise generated by the random measure M. Multiple stochastic
integrals form non-linear filters. Multiple integrals with respect to a Gaussian
measure were defined by Norbert Wiener and Kiyosi It6 in the 1940s. Their work
is recorded in Wiener (1958) and It6 (1951) respectively. Whereas the closure
of the linear span of these integrals equals the L2 space generated by the
Gaussian measure, no analogous result holds in the a-stable case, 0 < a < 2. The
construction of multiple integrals with respect to an a-stable measure with index
a < 2 is thus more difficult. Sufficient conditions for integrability are obtained by
Surgailis (1981, 1985) for 1 < a < 2 and by Samorodnitsky and Szulga (1989)
for 0 < a < 2. Kallenberg and Szulga (1989) give conditions for the integrability
of iterated Poisson integrals. McConnell and Taqqu (1986) show that the study of
double integrals with respect to a stable measure can be reduced to the study of
quadratic forms in independent stable variables. Explicit necessary and sufficient
conditions were developed by Rosinski and Woyczyrtski (1986) for double
integrals in the case 1 < a < 2 and extended by Kwapieii and Woyczynski (1987) to
0 < a < 2. McConnell (1986) extends these conditions to triple integrals when
1 < a < 2. For a summary, see Kwapieii and Woyczynski (1992). Extensions
to Banach-valued integrands are considered by Krakowiak and Szulga (1988) and
Samorodnitsky and Taqqu (1991b). For tails of multiple integrals with respect
to a SaS random measure 0 < a < 2, see Samorodnitsky and Szulga (1989)
in the case of real-valued integrands and Samorodnitsky and Taqqu (1990c) for
integrands taking value in a Banach space.
Section 3.5. The result of Theorem 3.5.3 was first noticed (in the symmetric case)
by Schilder (1970); his proof is incorrect in the case 1 < a < 2. The correct
proof has been presented by many authors. See, e.g., Hardin (1984) for the proof

578 HISTORICAL NOTES AND EXTENSIONS 14.4
in the general (skewed) case.
Part (ii) of the representation theorem (Theorem 3.5.6) is due to Schilder
(1970), and Part (iii) to Hardin (1984). We present a unified proof in Chapter 13.
Section 3.6. For a discussion of the Omstein-Uhlenbeck and reversed Omstein-
Uhlenbeck SaS processes and for an analysis of the Markov property in stable
processes, see Adler, Cambanis and Samorodnitsky (1990).
Examples 3.6.5 and 3.6.6 are important examples of a-stable self-similar
processes and are discussed in detail in Chapter 7. The well-balanced symmetric
linear fractional stable motion was introduced by Taqqu and Wolpert (1983)
and Maejima (1983ft). Fractional Brownian motion in the present terminology
goes back to Mandelbrot and Van Ness (1968), but a Gaussian process with
autocovariance function
R(t,s) = C{\t\a + \s\a-\t-s\a}
was mentioned in Kolmogorov (1940). (For more details, refer to the notes on
Chapter 7.)
Sections 3.7 and 3.8. Scale mixing of stochastic processes means randomizing
their scale parameter. Scale-mixed Gaussian and scale-mixed SaS processes are
"sub-Gaussian" and "sub-stable" processes, respectively. Their study goes back
to Bretagnolle, Dacunha-Castelle and Krivine (1966).
The integral representation of sub-Gaussian and sub-stable SaS processes
given in Propositions 3.7.1 and 3.8.2 is due to Hardin (1982ft).
Section 3.11. The idea of using the conditional normality of SaS processes in
order to tap the wealth of information available on the Gaussian processes goes
back to Marcus and Pisier (1984). This method has been widely applied ever since.
The proof that all SaS processes satisfying Condition S admit the representation
(3.11.4) is given in Chapter 13.
Section 3.12. The representation of stable processes as Poisson integrals is
sometimes referred to as the L6vy-Ito representation; see It6 (1969).
14.4 Notes to Chapter 4
Section 4.1. Kanter (1972) was the first to notice that the regression E(X2\Xi)
is linear when the vector (X\,X2) is SaS with 1 < a < 2. The simpler proof
given here is due to Linde (1986), Lemma 7.1.3. Fix (1949) established linearity
in a particular case.

14.4
HISTORICAL NOTES AND EXTENSIONS
579
Corollary 4.1.3 on multiple regression is due to Miller (1978). Miller gives
also necessary and sufficient conditions for the linearity of the multiple regression
in the SaS case. These are stated in Exercise 4.6 but arc, in general, not easy to
verify.
Corollary 4.1.5 treats the particular case of conditioning on independent
random variables. It is due to Kanter (1972). Parts (i) and (ii) of Proposition 4.1.7
which identify the sub-Gaussian SaS random vectors with 1 < a < 2 as having
the multiple regression property are due to Hardin (1982a).
Section 4.2. Tortrat (1975) claimed, mistakenly, that for any SaS random vector
(Xx, X2) with 1 < a < 2 the conditional law of X2 given X\ is symmetric
around the conditional mean. Linde and Mathe (1983) showed that this is wrong
and derived the necessary and sufficient conditions for symmetry presented in this
section.
Section 4.3. The results of this section are due to Adler, Cambanis and Samorod-
nitsky(1990).
Section 4.4. The asymptotic behavior of the distribution tails of order statistics in
an a-stable sample given in Theorem 4.4.5 was derived by Samorodnitsky (1988).
Section 4.5. Lemma 4.5.2 in the SaS case is due to Miller (1978).
Theorem 4.5.6 which presents necessary and sufficient conditions for the existence of
joint moments of a-stable random variables is due to Samorodnitsky and Taqqu
(19906).
Section 4.6. See Esary, Proschan and Walkup (1967) for basic properties of
associated random variables. Pitt (1982) proved that jointly normal random
variables are associated if and only if their correlations are all non-negative (see
also Joag-dev, Perlman and Pitt (1983)). The results on association of stable
random variables given here are due to Lee, Rachev and Samorodnitsky (1990).
For limit theorems, see Dabrowski and Jakubowski (1993).
Slepian inequalities provide ways to compare distributions of random vectors.
Slepian inequalities for SaS (and infinitely divisible) random vectors are given
in Samorodnitsky and Taqqu (19926,1994).
Section 4.7. The evaluation of the codifference and, more generally, of the
function U defined in (4.7.4) for Omstein-Uhlenbeck processes (Example 4.7.1)
comes from Levy and Taqqu (1991). That paper also contains Theorem 4.7.3 (the
moving averages case) and Proposition 4.7.4 (the sub-Gaussian case).
The behavior of the codifference for a stationary SaS process is closely related
to the mixing properties of the process where "mixing" is in the sense of ergodic

580 HISTORICAL NOTES AND EXTENSIONS 14.4
theory. Here is a brief overview.
Let Q. = RT and Tx be the cr-algebra generated by the stationary process
{X(t), t £ T} denned by X(t,uj) = w(t). For convenience, we assume T
is either R or E+. Let Ss : Q —* Q. be a shift transformation defined by
(5sw)(i) = u(t + s). The family {Sa}3€r is a semi-group of measure
preserving transformations on Q. Ergodicity, weak mixing and mixing of the process
{X(t), t € T} can now be defined in terms of {5s}ser.
• Ergodicity or metric transitivity:
(a) If Ss (A) = A for all s € T, then P(A) = 1 or P(A) = 0
(i.e., {Ss} has no disjoint components) or, equivalently,
1 /"*
lim-/ P{Ss{A)nB)ds = P{A)P{B).
(b)
Jo
• Weak mixing:
c
1 /■'
a) lim-/ |P(S.(i4)nB)-P(i4)P(B)|2ds = 0
or, equivalently,
(b) lim P(S,(A)nB) = P(il)P(B)l
where E is a subset of T with density 1, i.e., limt—oo i-1 |E fl [0, t]| =
lifT = R+.
• Mixing:
lim P{SS{A) r\B) = P{A)P(B).
s—»oo
The following implications hold for any stationary process:
mixing =>• weak mixing =>• ergodicity.
For stationary SaS processes, the following holds:
• Ergodicity and weak mixing coincide (Kokoszka & Podg6rski 1993)).
• Moving averages are mixing (Cambanis, Hardin, Jr. &Weron 1987).
• Sub-Gaussian and real harmonizable SaS processes are not ergodic
(Cambanis et al. 1987). This fact, in the harmonizable case, also follows from
Maruyama (1970).
• A necessary and sufficient condition for the process to be mixing is
lim Uifiu02;t)=0 forall0i,02eR (14.4.1)
t—*oo

14.4
HISTORICAL NOTES AND EXTENSIONS
581
This result, which is valid for any stationary infinitely divisible process, is
due to Maruyama (1970). For stable processes, by (4.7.6), Relation (14.4.1)
is equivalent to
lim 1(0, ,62;t)=0 for all 9X, 92 e R.
t—>oo
This, in fact, provides an alternate proof of Theorem 4.7.3 since, as noted
above, moving average SaS processes are mixing and r(t) = —I{— 1,1, t).
Maruyama's result requires I{9\ ,92;t) —* 0 as t —» oo for all 9\, 92 € R in order
to ensure that the process is mixing. Gross (1993) proves that it is not necessary
to require I(9\ ,92;t) —> 0 for all 9X, 92 € R if the process is SaS and satisfies
Condition S (see Section 3.11). It is only necessary to choose for {9\,92) the
value (-1,1) if 0 < a < 1 and the values (-1,1) and (1,1) if 1 < a < 2. Since
I(9i,62;t) —> 0 is equivalent to U(9\,92;t) —> 0, this can also be expressed as
U(-l,l;t) ->0when0<a < 1 and U{-1, l;t) -> 0 and 17(1,1; i)-> 0 when
1 < a < 2.
An extension of Gross' result to processes of type G can be found in Kokoszka
and Taqqu (1993b). Processes of type G extend SaS processes. They are
infinitely divisible processes which admit, conditionally, a series representation.
In particular, a random variable X is of type G if and only if X = ZS, where Z
has the N(0,1) distribution and 5 is a nonnegative random variable independent
of Z such that S2 is infinitely divisible. (Informally, X ~ N{0,S2).) For a
comprehensive account of distributions and processes of type G, refer to Rosinski
(1990, 1991).
A stationary Gaussian process is ergodic if and only if its spectral measure
(the Fourier transform of the autocovariance function) has no atoms; see, e.g.,
Rozanov (1967). What about SaS processes?
The shift Ss induces a shift Ss on random variables as follows: define firstly
Ss on indicator functions by Ss\a — 1s_„a, A € Fx, and then extend it by
linearity and continuity in probability to all random variables measurable with
respect to Tx- Let {X(t), t e T} be a SaS process with index 0 < a < 2.
Consider £(JFX), the space of all SaS random variables measurable with respect
to Tx- If Y is such a random variable, then {SsY}sST is a ^stationary SaS
stochastic process. For example, if Y = 2X(1) + 3X(2), then SSY = 2X(1 +
s) + 3X(2 + s), s € T. Conditions for ergodicity can be formulated in terms of
C{TX).
Cambanis, Hardin and Weron (1987) prove that a stationary SaS processes
{X(i}}, 0 < a < 2, is ergodic if and only if for each random variable Y 6
lim i [t\\S,Y-Y\\aads = 2\\Y\\aa
'-*00 t Jo

582 HISTORICAL NOTES AND EXTENSIONS 14.5
and
lim j fwSsY-Ylfcds^AWYWl*.
Podg6rski (1992) gives a different characterization. He proves that the stationary
SaS process X is ergodic if and only if for each Y € C{Tx),
Urn^ i J exp(2||y||S - \\S.Y - Y\\°)ds = 1. (14.4.2)
This relation can be rewritten succinctly in terms of the codifference ty (s) of the
stationary process {Ssy}ser as
1 /*'
lim - / eTY^ds= 1 (14.4.3)
t-~oo t J0
or, in terms of the corresponding function Uy, as
1 f*
lim - / UY(l,-l;s)ds = 0. (14.4.4)
t —00 t J0
Section 4.8. The first results concerning crossings of stationary Gaussian
processes were obtained by Rice (1939, 1944, 1945). More rigorous proofs were
given by several authors and can be found in Cramer and Leadbetter (1967) and
Leadbetter, Lindgren and Rootz£n (1983). See these books for a detailed history.
Adler, Samorodnitsky and Gadrich (1993) investigate the asymptotic behavior of
the mean number ECU of crossings of a level u, as u —> oo, in the case where the
process X is SaS and stationary. When X is sub-Gaussian, they obtain the result
of Theorem 4.8.2. They also consider the case where X is a real harmonizable
SaS (as defined in Example 3.6.7). Using the fact that X is a conditionally
stationary centered Gaussian process (Proposition 6.6.4), they prove that ECo < oo
implies
ECu ~ ^u-a
as u —► co. The constant Ca (do not confuse it with Cu!) is defined in (1.2.9) and
Ai is the first spectral moment of the process Gaussian G in (6.6.4). Thus, as in
the sub-Gaussian case, ECU is asymptotically proportional to u~a. This problem
for other SaS processes, e.g., moving averages, is still open.
14.5 Notes to Chapter 5
Section 5.1 Theorem 5.1.2, which relates the finiteness of moments to the behavior
of the characteristic function at the origin, is due to Ramachandran (1969); see

14.5
HISTORICAL NOTES AND EXTENSIONS
583
also Ramachandran and Rao (1968). Our formulation is slightly different from
the one stated in Ramachandran (1969).
The relation between the integrability properties of the spectral measure of a
SaS random vector and the existence of its conditional moments was discovered
by Samorodnitsky and Taqqu (1991a). That paper contains the proof of Theorem
5.1.3 in the symmetric case for 0 < p < min{o: + v,2}. The corresponding
result in the skewed case and the form of the regression was obtained by Hardin,
Samorodnitsky and Taqqu (1991a). The extension to 0 < p < min{a+v, 2a+l}
is due to Cioczek-Georges and Taqqu (1994a, 1994c).
LePage (1990a) used an alternative approach, taking advantage of the
conditional normality of SaS random vectors. While such an approach appears to be
more restrictive than the one presented in this chapter, it can be used in some cases
to prove the existence of absolute conditional moments of order greater or equal
to a.
How sharp is Theorem 5.1.3? If one defines ■E(|X2|p|.X'i = x) as the pth
moment of the distribution with characteristic function (5.1.9), then E{X%\Xi =
0) < co is actually equivalent to (5.1.10) with v = 2-a (Wu & Cambanis 1991).
Cioczek-Georges and Taqqu (1994b) extend this result to other values of p. They
show under the same conditions that U(|.X2|p|-^i = 0) < oo is equivalent to
(5.1.10) with v = p — a for
a<p<2a+l if 0 < a < 1/2 or 1 < a < 3/2,
a <p< 2 if 1/2 <a < 1,
a < p < 4 if 3/2 < a < 2.
Moreover, in the case p = a,
E(\X2\p\Xi = 0) < oo ifandonlyif - / ln|si|r(ds) < co.
Js2
One can extend Theorem 5.1.3 to the case of conditional moments of the
type E{\Xd\p\Xi = xu...,Xd-i — x<i-i) for an a-stable random vector
(X[,..., Xd_i, Xd) with d > 3 if one replaces the integrability condition (5.1.10)
by the condition
P(*l,.-.,*--!)= / ,, q . T{dl „ |, < °° t14-5'1)
Jsd 1*1^1 H rid-lSd-il"
for some v > 0, for almost all (t\,...,ta-\) € Sd-i, and imposes in addition
certain integrability conditions on P(t\,... ,*d-i) overKd_I. Cambanis andWu
(1992), for example, show that in the case 0 < a < 1, the multiple regression

584 . HISTORICAL NOTES AND EXTENSIONS 14.5
E{\Xd\\Xl=xl,...,Xd-l
a v > 1 - a such that
f |toc(ti,...,*d-i,0)| /
id_i) exists for almost all x\,..., xd if there is
r(rfs)
r- ■ — r-dt\ ...dtd-i < oo.
isd l*isi +--- + id-iSd-ir
Unfortunately, in many cases of interest, either (14.5.1) does not hold for large
enough vs or P(t\,..., td-i) does not possess the required integrability properties.
Nevertheless, the integrability condition (5.1.9) for d = 2 is still useful when d > 2
because
E(\Xd\"\Xi) <ooa.s. => E(\Xd\P\Xu...,Xd-.l)<ooiLs.
Section 5.2. In view of the results of Cioczek-Georges and Taqqu (19946,1994a,
1994c), Condition (5.2.4) can be replaced by the weaker condition:
Jst
'Is
s,|Q_T(ds) < oo-for a < 1,
In \s\ \T(ds) < oo for a = 1.
s2
(See also the remarks following Theorem 5.1.3.) The form of the non-linear
regression and the necessary and sufficient condition for linearity of regression
presented in this section are due to Hardin, Samorodnitsky and Taqqu (1991a).
Wu and Cambanis (1991) provide further insight into the structure of
conditional moments of stable laws by computing the conditional variance.
Specifically, let {X], X2) be SaS, 1 < a < 2, with spectral measure T and suppose that
(5.1.10) holds with i/ = 2-a. Then for a.e. x 6 E,
Var (X2\Xi =x)= C(a,T)h(a, —V (14.5.2)
where
C(a,r) = er
Is
*i
|a-2e2
S2r(ds)
h(a,x) = / ufa{u)du
/qW J\x\
and where k is defined in (5.2.10) and fa is the density function of the Sa (1,0,0)
law.
Cioczek-Georges and Taqqu (19936) extend the preceding result to the case
1/2 < a < 1. They show that if (5.1.10) holds with v > 2 - a for 1/2 < a < 1
or v - 2 - a for 1 < a < 2, then (14.5.2) holds with
_,2 roo
h(a,x) = ——-r costxe-tat2a-2dt + x2.

14.6
HISTORICAL NOTES AND EXTENSIONS
585
Observe the separation of factors in (14.5.2): the constant C(a, F) depends on the
joint law of (X\, X2) but the function h(a, x) involves only the density of X\.
The paper (Cioczek-Georges & Taqqu 1993a) contains necessary and
sufficient conditions for asymptotic linearity of E[Y\X + £ = z] where (X, Y) is an
a-stable random vector and £ is a random variable, independent of (X, Y), such
that X + £ is in the domain of normal attraction of X. Asymptotic linearity does
not always hold even when .E[Y|X = x] is linear. For some distributions of £,
the asymptotic rate of ^[VIX + £ = z] fluctuates.
Cambanis and Fakhre-Zakeri (1993) study the prediction of heavy-tailed
AR(1) sequences. The innovations are assumed semistable, i.e. their
characteristic function satisfies
<p{r) = <pT {\r)ei,ir for some 0 < |A| < r, /i € R, and for all r£l.
Semistable laws which were introduced by Levy (1954), p. 203 generalize the
stable laws.
Section 5.3. Sub-Gaussian and harmonizable vectors are SaS random vectors
whose conditional moments can be studied by the methods of LePage (1990a).
Using the comments following the proof of Corollary 5.3.3 (or the necessary
conditions in Cioczek-Georges and Taqqu (1994£>)), part (i) of that corollary can
be strengthened as follows:
If X = (X\, X2) is sub-Gaussian a-stable and X\ and X2 are linearly
independent, then
E(\X2\V\X\ — x) < ooa.e. if and only if p<a+l-
Section 5.4. The regression graphs of Section 5.4 are from Hardin, Samorodnitsky
and Taqqu (1991a).
Section 5.5. The paper Hardin, Samorodnitsky and Taqqu (1991b) describes
the numerical techniques and lists the source code of the software package for
computing the non-linear regressions.
14.6 Notes to Chapter 6
Section 6.5. Marcus and Pisier (1984) refer to stationary SaS processes of the
form (6.5.1) as strongly stationary.
Section 6.6. The fact that stationary harmonizable SaS processes are
conditionally stationary Gaussian was discovered by Marcus and Pisier (1984).

586 HISTORICAL NOTES AND EXTENSIONS 14.7
The relation between the classes of stationary harmonizable SaS processes
and stationary sub-Gaussian SaS processes was first discussed in Cambanis and
Soltani (1984).
14.7 Notes to Chapter 7
Section 7.1. Processes that are self-similar with index H and have stationary
increments (iJ-sssi processes) are sometimes called "statistically self-similar" in
order to distinguish them from non-random fractals that are self-similar.
Sometimes the term "self-affine" or "statistically self-affine" is used.
Benoit Mandelbrot's pioneering work on self-similarity dates to the early
1960s, e.g., Mandelbrot (1965). John Lamperti, in his seminal paper (1962),
showed that self-similar process result from limits of normalized sums. Lamperti
used the term "semi-stable" and allowed for shifts.
It is now common to use the letter H to denote the index of self-similarity. H
is the first letter of the British Harold Edwin Hurst's last name, who spent 62 years
in Egypt, mostly working on projects involving the river Nile. Using the "range"
statistic, he discovered empirically that cumulative yearly flows of the Nile obey
a power law with H = 0.7 instead of the expected H = 0.5 (Hurst 1951). His
work was the basis of Mandelbrot's suggestion that fractional Brownian motion
be used to model the yearly levels of the river (see Mandelbrot (1965)).
Proposition 7.1.4 which relates an if-sssi process to a stationary one is due to
Lamperti (1962). The proof of Lemma 7.1.9 is adapted from O'Brien and Vervaat
(1983) and that of Proposition 7.1.10 comes from Maejima (1986).
Brownian motion, the oldest example of an iJ-sssi process, is named after
the biologist Robert Brown whose research dates to the 1820s. Early in this
century, Louis Bachelier, Albert Einstein and Norbert Wiener began developing
the mathematical theory of Brownian motion. The construction of Bachelier
(1900) was erroneous but it captured many of the essential properties of the
process. It is worth reading the short but fascinating account of Bachelier's
tribulations in Mandelbrot (1982).
Section 7.2. Andrei Nikolaevich Kolmogorov discovered fractional Brownian
motion. The article, written in German, appeared in 1940 in the Soviet "Dokladi."
The process is introduced in Kolmogorov (1940) in the context of transformations
of curves in Hilbert space. It was then briefly studied by Hunt (1951) and set
in a wider framework by Akiva M. Yaglom (1955). It is, however, the seminal
paper of Mandelbrot and Van Ness (1968) that investigated the basic properties
of fractional Brownian motion and emphasized its relevance to the modeling of
natural phenomena. The following papers, for example, Mandelbrot and Wallis

14.7
HISTORICAL NOTES AND EXTENSIONS
587
(1969a, 1969c, 19696), develop applications to hydrology.
The moving average representation (7.2.4) of fractional Brownian motion can
be found in Mandelbrot and Van Ness (1968). It involves a fractional-type integral
used earlier in a non-stochastic context; see, e.g., Zygmund (1979). Kolmogorov
(1940) and Yaglom (1955) used the harmonizable representation (7.2.12). A more
general version of Proposition 7.2.7 which relates the two representations can be
found in Taqqu (1979).
The terms fractional Brownian motion and fractional Gaussian noise were
coined by Mandelbrot. Fractional Gaussian noise can be defined as the increment
of a Gaussian if-sssi process or, directly, as a fixed point of the renormalization
transformation. Sinai (1976) takes this second approach and derives its
autocovanance (7.2.21) and spectral density (7.2.22). Cox (1984) advocates a similar
point of view.
In order to gain modeling flexibility, it is important in applications to be
able to replace fractional Gaussian noise by an arbitrary Gaussian moving
average whose autocovariance function has the same asymptotic behavior as that of
fractional Gaussian noise. Such moving averages are said to belong to the
"domain of attraction" of fractional Brownian motion. More generally, a stationary
sequence {Yj, j = 1,2,...} belongs to the domain of attraction of a process
X = {X(t), t > 0} if {£j=i *ii * ^ 0} suitably normalized, converges as
JV —> co to the finite-dimensional distribution of X. Theorem 7.2.11 (Taqqu
1975) provides a simple example of a moving average in the domain of attraction
of fractional Brownian motion. Other examples can be found in Davydov (1970),
Rosenblatt (1961), Taqqu (1979, 1981), Dobrushin and Major (1979) and Major
(1981).
There are a number of papers involving statistical estimation. See, e.g.,
Mandelbrot and Taqqu (1979), Fox and Taqqu (1986), Dahlhaus (19.89) and
Robinson (1991, 1992a, 19926). Beran (1992) provides a good overview. For
examples of statistical analysis, see Beran, Sherman, Taqqu and Willinger (1993)
and Leland, Taqqu, Willinger and Wilson (1993, 1994). Samarov and Taqqu
(1988) and Yajima (1988, 1989) study regression models with Gaussian errors
that have long-range dependence.
Section 7.4. The linear fractional stable motion (7.4.1) was introduced by Taqqu
and Wolpert (1983) as an integral with respect to a Poisson measure and by
Maejima (1983c) as an integral with respect to a stable measure. Astrauskas (1982)
provides a generalized field version. Theorem 7.4.5 was proved by Cambanis and
Maejima (1989) in the case 1 < a < 2 and j3 = 0 and by Samorodnitsky and
Taqqu (1989) in the general case.
In the past, a "stable process" meant a-stable L£vy motion. This usage is

588 HISTORICAL NOTES AND EXTENSIONS 14.7
confusing and should be discontinued because one needs to distinguish between a
process with independent increments and the more general stable processes whose
increments may be dependent. The importance of a-stable LeVy motion stems
from the fact that it is the stable analog of Brownian motion. Theorem 7.5.4,
which states that it is the unique 1/a-sssi process when 0 < a < 1, is due to
Samorodnitsky and Taqqu (1990a). The log-fractional stable motion, an 1/a-
sssi process, 1 < a < 2, different from a-stable LeVy motion, was discovered by
Kasahara, Maejima and Vervaat (1988). Other 1/a-sssi processes with 1 < a < 2
can be found in Samorodnitsky and Taqqu (1990a) (see Exercises 7.10 and 7.12).
Sections 7.7 and 7.8. The real and complex harmonizable fractional stable
motions were considered by Cambanis and Maejima (1989). Theorem 7.8.2 is
proved in that paper in the case 1 < a < 2, but the identical proof extends to
0 < a < 2 once one notices that the process in (7.8.6) is well defined. A good
review of a-stable Jf-sssi processes is given by K6no and Maejima (1991).
Section 7.10. Linear fractional stable noise is the increment of linear fractional
stable motion. Theorem 7.10.1 concerns the asymptotic behavior of the function
1 and the codifference for linear fractional stable noise. Its proof was sketched by
Astrauskas (1983) in the SaS case. The proof in the general a-stable case can be
found in Astrauskas, Levy and Taqqu (1991).
Section 7.11. Fractional Gaussian noise has different representations and any one
of them can serve as a basis for an "approximate" simulation procedure. Here, we
have adapted the "Type 1" approximation of Mandelbrot and Wallis (1969a) and
applied it to the case of SaS innovations as well. Mandelbrot and Wallis's rougher
approximation, called "Type 2", consists of simulating Yjj^XfU ~ u)d~leu,
j > 1, where M is a large number, d = H — 1/a, and the eus are the
innovations.
There are also "fast" simulation methods of fractional Gaussian noise. (See
for example Mandelbrot (1971) and Granger(1980).) But since that process is
Gaussian, it is also possible to use the following "exact" simulation method.
Let Ct = (r(i — j)), i,j = 1,.. .T be the covariance matrix of Yi,... ,Yt
defined in (7.2.21) with r0 = 1. Performing a Cholesky decomposition on Ct,
yields Ct = MM', where M is a T x T lower trianguluar matrix with entries
mju, j,u = l,...T and where M' denotes the transpose of M. Then, if
eu, u= l,...,Tarei.i.d. 7V(0,1),
T
u=l
is fractional Gaussian noise with variance a\. This method, which is only practical

14.7
HISTORICAL NOTES AND EXTENSIONS
589
for moderate values of T (e.g. T = 200) is described in McLeod and Hipel (1978).
Section 7.12. There is a large literature on ARMA time series with finite variance.
Since it is essentially an L2- theory, there is often no loss of generality in supposing
the innovations Gaussian. Brockwell and Davis (1991) offer a good exposition
and also provide a number of results in the case of SaS innovations. Cline (1983),
Cline and Brockwell (1985) and Davis and Resnick (1986) consider this latter case
as well. Kokoszka and Taqqu (1993d) show that the codifference of ARMA time
series with stable innovations behaves like the covariance in the Gaussian case: it
fluctuates within a band of two exponentially decreasing functions.
A number of statistical results are also available for moving averages Xn of
the form (7.12.3). For instance, Kluppelberg and Mikosch (1993c) obtain the
limit of the normalized periodogram In.xW = \N~x/a J2n=i Xne~inX\2 as
N —> oo, when the innovations are in the domain of normal attraction of a SaS
distribution, 0 < a < 2. For fixed A = 2itlj, lj G (0,1/2) irrational, the limit is
|^(A)|2 (a2(A) + /32(A)) where ^(A) = Ejl-oo ej e~XJ (cj ^the coefficients
of the moving average) and (a(A),/3(A)) = {All2G\,Axl2G-i) is sub-Gaussian.
In contrast to the Gaussian case, for 0 < A < ... < Am < 7r, the periodogram
ordinates Ijv,x(Ai), i = 1,... ,m, are not asymptotically independent.
Davis and Resnick (1986) derive weak limits of sample autocovariance and
autocorrelation functions of Xn under the assumption that the distribution of
the innovations has tails P(|ei| > x) = x~aL{x), where L is a slowly
varying function. Their results have been generalized by Kluppelberg and Mikosch
(1993b) who also consider the convergence of In,x(A) = In,xW/Yln=i ^n as
N —» oo. Kluppelberg and Mikosch (1993a), study the behavior of the integrated
periodogram and apply the results to goodness-of-fit tests.
Estimators for the parameters of the AR(p) model with stable innovations
or with innovations in the domain of attraction of a stable distribution have
been studied by various authors. Yule-Walker, least square and least deviation
estimators (the latter minimizes £]„_., l-^™ ~ 4>\Xn-\ - •■■ - <ppXn-p\) have
been considered. See Kanter and Steiger (1974), Hannan and Kanter (1977),
Gross and Steiger (1979), An and Chen (1982), Davis and Resnick (1986), Knight
(1986, 1987), Liu (1987), and Davis, Knight and Liu (1992).
Mikosch, Gadrich, Kluppelberg, and Adler (1993) also consider the estimation
of the parameter vector (<£],..., <j>p, 9\,..., 6q) of an ARMA (p, q) sequence
whose innovations e„ satisfy the aforementioned conditions and, in the case of
SaS innovations, they study a sample periodogram based estimator of the vector
(<pu...,<t>p,9\,...,9q). Bhansali (1993) estimates the coefficients of (7.12.3)
when an autoregressive representation exists and the tjS are in the domain of
attraction of a SaS law. All these papers assume V. \cj\ < oo or impose

590 HISTORICAL NOTES AND EXTENSIONS 14.8
even stricter summability conditions on the sequence {cj} in order to ensure the
absolute a.s. convergence of the random series (7.12.3).
Section 7.13. Fractional ARIMA time series extend the realm of ARMA modeling
to include long-range dependence. Brockwell and Davis (1991) devote a section
of their book to fractional ARIMA with Gaussian innovation. The model was
considered by Adenstedt (1974) and developed by Granger and Joyeux (1980) and
Hosking (1981). See Samorodnitsky and Taqqu (1992a) for a brief description.
Kokoszka and Taqqu (1993c) develop the theory of fractional ARIMA when
the innovations are SaS, 0 < a < 2, and evaluate the codifference. The proofs
of Theorems 7.13.3, 7.13.4, 7.13.5 and Exercise 7.18 can be found in that paper.
Kokoszka and Taqqu (1993e) extend the results to moving averages with regularly
varying coefficients.
Fractional ARIMA is of interest because it presents an alternative way to model
long-range dependence. In fact, any stationary sequence of random variables
whose normalized partial sums converge weakly to linear fractional stable motion
would do as well. See Avram and Taqqu (19866) and Maejima (1989).
14.8 Notes to Chapter 8
The Ldvy Brownian motion (Example 8.3.3) is the best known example of an
H-sssis Gaussian random field. Paul L6vy (1965) devotes a chapter to its study.
Chentsov (1957), in a beautiful short note gives the geometric construction
presented in Section 8.3 with Vt equal to the set of hyperplanes separating the origin
from the point t. There is also an interesting discussion in L6vy (1966). McKean
(1963) and Cartier (1971) show that the field has a kind of Markovian property if
the parameter space is Kn with n odd.
While LeVy Brownian motion is jEf-sssis with H = 1/2, the L6vy fractional
Brownian field (Example 8.1.3) is ff-sssis for all H e (0,1). The LeVy fractional
Brownian field was first considered by Yaglom (1957) who characterized its
spectral representation. It was then investigated by Gangolli (1967) and Mandelbrot
(1975, 1982) among others. Mandelbrot's paper contains the germs of future
developments.
Shigeo Takenaka (1987) gives a geometric construction for the L6vy fractional
Brownian field with 0 < H < 1/2, using the Vt described in Section 8.4.
In Takenaka (1991), he defines the Takenaka and (a, #)-Takenaka fields (he
calls them "of Chentsov type"), by considering SaS, 0 < a < 2, random
measures instead of merely Gaussian measures. He also introduces the Chentsov
random fields which he calls "generalized Chentsov." Mori (1992) gives a general
characterization of the L6vy-Chentsov random field in the context of infinitely

14.8
HISTORICAL NOTES AND EXTENSIONS
591
divisible measures.
The proof of Theorem 8.2.6, which gives sufficient conditions for a Chentsov
field to be ff-sssis, is due to Cioczek-Georges. An independent proof can be
found in Takenaka (1993). The results of Sections 8.5 and 8.6 are due to Yumiko
Sato and Shigeo Takenaka (see Sato and Takenaka (1991) and Sato (1992a)).
Results concerning the codifference and the function I{6\, #2; u) in Section 8.7
were obtained by Kokoszka and Taqqu (1992).1
The extension of Theorem 8.8.3 to R1 and R2 can be found in Sato (1992ft).
The extension to M.n (Theorem 8.8.4) is proved in Sato (1991) by relating it to the
intersection properties of a family of cones in W1 x R+.
This chapter considered only fields that are H-sssis. If one replaces the H-sssis
requirement by if-sssi, then one can include many other random fields. Kokoszka
and Taqqu (1993/) introduce a large class of such fields, defined by
xv(t) = f (p(* - x)"~n/a - p(x)B-n/a)M{dx), t e Rn, (14.8.1)
JM.n
where p is a norm on W1 and M is a SaS random measure on Rn with Lebesgue
control measure. The field Xp is well defined for any n > 1, H € (0,1), a e
(0,2] and is ff-sssi. If n = 1, it reduces to the well-balanced linear fractional
stable motion (Example 3.6.5). The family (14.8.1) parametrized by p is extremely
rich: if two fields Xp and Xp< have identical finite-dimensional distribution, then
the norms p and p' must be identical. The special choice
P = PaAx) = (J \{x, s)\ra(ds)\ , x 6 W1,
where r > 1, Sn is the unit sphere in Mn and a is a symmetric finite positive
Borel measure on Sn, gives rise to a large class of interesting models. Kokoszka
and Taqqu show in (1993a) that, for any choice of norm p, the codifference t(u)
related to Xp is asymptotically proportional to uaiI~a as u —> oo if n > 1.
Since the codifference related to Chentsov fields is asymptotically proportional to
uaH~2 (Theorem 8.7.1), Chentsov fields are different from the fields Xp.
Surgailis, Rosinski, Mandrekar and Cambanis (1993) define mixtures of fields
which they call "generalized moving averages." These fields can be represented
as
X(t)= J f(X,t-x)M{dX,dx), teW1,
JAxR"
where M is a SaS random measure onAxl" with control measure Q<8>Leb,
i.e., the control measure of M is the product measure of a measure Q on A and
'The proof of Theorem 3.1 in Kokoszka and Taqqu (1992) is incorrect because increment processes
involve three- and not two-dimensional distributions. Whether the theorem itself holds is still an open
problem.

592 HISTORICAL NOTES AND EXTENSIONS 14.10
Lebesgue measure on Rn. The space (A, Q) characterizing the mixture is assumed
to be a cr-finite complete measure space and the kernel /: A x Kra —► R is assumed
measurable and such that
f \f(\,x)\aQ(d\)dx <oo.
JAx&n
If Q has only one atom or if the space A consists of only one point, then the field
X becomes the usual SaS moving average. If Q has two atoms, then X is a
mixture of two SaS moving averages. Observe that Takenaka fields are mixtures
with A = E+ and Q — mp.
14.9 Notes to Chapter 9
Section 9.2. Stochastic processes defined on uncountable parameter spaces give
rise to measurability problems. The idea of using separability to overcome these
problems goes back to Doob (1953). We shall use his approach for constructing
a separable version of a stochastic process.
Section 9.5. Zero-one laws for a-stable processes with 0 < a < 2 were first
discussed by Dudley and Kanter (1974). The approach we present here is new.
14.10 Notes to Chapter 10
Section 10.2. De Acosta (1975) observed that the tail of the distribution of a
measurable semi-norm of an a-stable random vector, 0 < a < 2, is dominated
by the tail of the absolute value of a one-dimensional a-stable random variable
(our Proposition 10.2.1 is a particular case of this statement). The following more
precise version of the former result was proved in de Acosta (1977):
Let E be a real vector space, and B be a o-field generated by a linear vector,
space of linear functionals on E. A probability measure p. on (E,B) is called
a-stable, 0 < a < 2, if for any A,B>0, there isaD € E such that
AXi + BX2 = (Aa + Ba)l/aX + D,
where X], X? are independent copies o/X, all with law p.Ifq is a measurable
semi-norm on E, then the limit
I = lim AQ/x(x e E: q(x) > A)
exists, and I € [0, oo). (The limit I is positive under certain non-degeneracy
conditions.)

14.11
HISTORICAL NOTES AND EXTENSIONS
593
Kono and Maejima (1991) state erroneously that the Takenaka process in
Example 10.2.7 has bounded sample paths.
Section 10.3. Rosinski (1986) observed that the sample paths of an a-stable
process with 0 < a < 2 "inherit" the negative properties of a "typical" function
f(t, x), t€ T, in the integral representation (10.1.1).
Section 10.4. The fact that it is easy to derive necessary and sufficient conditions
for sample boundedness and continuity of a-stable processes in the case 0 < a < 1
but not in the case 1 < a < 2 is closely related to the following fact from the
theory of probability on Banach spaces: it is easy to describe all Banach space-
valued a-stable random variables as long as the Banach space has the so-called
stable type p > a; see Linde (1986). Whereas any Banach space has stable type
p for any 0 < p < 1, many important ones do not have a stable type p > 1, e.g.,
spaces of bounded or continuous functions.
Sample boundedness and continuity of SaS moving averages with 0 < a < 1
was considered by Balkema and de Haan (1988).
Section 10.5. Theorem 10.5.1 is adapted from a more general result of Rosinski
and Samorodnitsky (1993).
Section 10.6. Oscillation processes were introduced by Ito and Nisio (1968)
for processes on the interval [0,1]. Jain and Kallianpur (1972) extended them to
stochastic processes indexed by a separable metric space and showed that Gaussian
processes that are continuous in probability have non-random oscillations.
Oscillation processes of non-Gaussian infinitely divisible processes were
considered by Cambanis, Nolan and Rosinski (1990). Noting that continuity in
probability is no longer the appropriate assumption, they introduced instead an
analogue of Condition 10.6.4.
14.11 Notes to Chapter 11
Section 11.1. The necessary and sufficient conditions that relate the integral
representation of an SaS process, 0 < a < 2, to existence of a measurable
version was discovered by Rosinski and Woyczyriski (1986).
Section 11.2. Conditions for integrability of the sample paths of a-stable processes
with 0 < a < 2 covering particular cases are scattered throughout the literature.
The (symmetric) case 1 < p < a was completely described by Cambanis and
Miller (1980) and Linde, Mandrekar and Weron (1980), whereas the case p >
max(a, 1) is due to Marcus and Woyczyriski (1979) and Linde, Mandrekar and

594
HISTORICAL NOTES AND EXTENSIONS
14.12
Weron (1980). The problem in the case p = a > 1 was solved by Rosinski and
Woyczyriski (1986).
As most of these results were obtained by working with a-stable measures on
Lp spaces, the case 0 < p < 1 was rarely considered. The case p = a e (0,1)
has been implicitly solved by Kwapiefi and Woyczyriski (1987), and sufficiency
of the integrability conditions in the case 0 < a < p < 1 can be deduced from
Marcus and Woyczyriski (1979) and Rosinski and Woyczyriski (1985). The unified
approach presented here is due to Samorodnitsky (1992).
Section 11.4. The problem of interchanging the order of ordinary and stochastic
integration in the SaS case with 1 < a < 2 was considered by Cambanis and
Miller (1981) and McConnell and Taqqu (1984), Lemma 4.4. Rosinski (1986)
gave a complete solution in the case 1 < a < 2. Our approach in the case
0 < a < 1 follows Samorodnitsky (1992).
Section 11.7. Cambanis and Miller (1980) derived necessary and sufficient
conditions for a SaS process {X(t), a < t < b} to have a version with all sample
paths in AC[[a, b}. Rosinski (1986) extended their result to the SaS case with
0<a < 2andp> 1.
14.12 Notes to Chapter 12
Section 12.1. The importance of metric entropy in the study of the sample path
properties of Gaussian processes has been underlined by Dudley (1967,1973) and
Fernique (1975), who established a variety of very precise results.
Pisier (1983) showed that one can derive sufficient conditions for sample
boundedness and continuity in a general non-Gaussian framework by using metric
entropy. Our approach follows that of Ledoux and Talagrand (1991).
Section 12.2. Another version of the sufficient condition for sample boundedness
and sample continuity of a-stable processes is given by Marcus and Pisier (1984).
See also Nolan (1989) for a more explicit form.
Section 12.3. The results of Theorem 12.3.1 are due to Marcus and Pisier (1984)
in the case 1 < a < 2 and to Talagrand (1988) in the case a == 1.
Section 12.4. The general exposition of this section as well as the breakdown of
the set V follows Kono and Maejima (1991).
Maejima (1983a) was the first to show that the well-balanced linear fractional
stable motions are not sample bounded on any integral of positive length if 0 <
H < l/a. Examples 12.4.2 and 12.4.3 are taken from Samorodnitsky (1993).

14.13
HISTORICAL NOTES AND EXTENSIONS
595
A more detailed discussion of sample path properties of certain a-stable self-
similar processes can be found in Takashima (1991) and K6no and Maejima
(1991). See also a general discussion in Vervaat (1985).
14.13 Notes to Chapter 13
Bretagnolle, Dacunha-Castelle and Krivine (1966) established the integral
representation for SaS processes in the case a > 1. They showed that for processes
continuous in probability, one can choose E =~ (0,1) in (13.2.4) and let M have
Lebesgue control measure. Schreiber (1972) extended their result to the case
0 < a < 1.
Schilder (1970), apparently unaware of Bretagnolle, Dacunha-Castelle and
Krivine's work, established the integral representation in the case 0 < a < 2
and T finite, noting that such a representation "for even countably many [time
points] xs has eluded the most vigorous efforts of the author." Kuelbs (1973)
reproved the result for countable T and extended it to the case of SaS processes
satisfying Condition S. Hardin (1984) then extended the integral representation to
the skewed a-stable processes with a ^ 1, satisfying Condition S.
The approach we follow here is close to the one in Ledoux and Talagrand
(1991) in the SaS case; it does not use the theory of infinitely divisible random
vectors in Banach space.

Appendix A
Tables of symmetric a-stable
fractiles
Let X ~ S(a, 0,0) and let / = P{X < x/} be the upper fractile corresponding
to xj. The following tables display x/ as a function of
/ = 0.500 (0.025) 0.900 (0.010) 0.970 (0.005) 0.995, 0.998.
for
a = 0.1 (0.1) 1.9,
and also as a function of the extreme upper fractiles1
/ = 0.9990 (0.0001) 0.9999 for a = 0.5 (0.1) 1.9.
Since X is symmetric, entries for / < 0.5 are obtained by using the relation
/ = P(X < xf) = 1 - P{X < xi-}).
These tables are reproduced from Brothers, DuMouchel and Paulson (1983), with
the kind permission of Albert Paulson. They are based, for the most part, on
Zolotarev's integral representation of the cumulative distribution function (see
Section 1.6). We refer the reader to the aforementioned paper for details on
the computational techniques. The listing for the symmetric Cauchy (a = 1) is
obtained by using the relation
x/ = tan7r(/-l/2),
which is aconsequence of (1.1.14). The listing for q = 2 is obtained by using the
N(0, v2) distribution, with v = y/l.
1 There are a few discrepancies with Paulson and Delehanty (1993) in the third and fourth significant
digits for / = 0.9998 and / = 0.9999.

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TABLES OF SYMMETRIC a-STABLE FRACTILES
599
/ 0.750
a = 2.0 .95387
a =1.9 .95680
a =1.8 .95976
a =1.7 .96274
a =1.6 .96577
a = 1.5 .96893
a =1.4 .97237
a =1.3 .97638
a = 1.2 .98154
a =1.1 .98885
a= 1.0 1.000
a = 0.9 1.0176
a = 0.8 1.0455
a = 0.7 1.0901
a = 0.6 1.1621
a = 0.5 1.2838
a = 0.4 1.5090
a = 0.3 2.0060
a = 0.2 3.6241
a = 0.1 22.255
/ 0.875
a = 2.0 1.6268
a =1.9 1.6476
a = 1.8 1.6724
a =1.7 1.7025
a = 1.6 1.7400
a =1.5 1.7878
a = 1.4 1.8502
a = 1.3 1.9333
a = 1.2 2.0458
a = 1.1 2.1998
a =1.0 2.4142
a = 0.9 2.7203
a = 0.8 3.1754
a = 0.7 3.8957
a = 0.6 5.1460
a = 0.5 7.6487
a = 0.4 13.976
a = 0.3 38.622
a = 0.2 300.84
a = 0.1 .14876 + 06
0.775 0.800
1.0683 1.1902
1.0729 1.1970
1.0777 1.2045
1.0830 1.2130
1.0888 1.2229
1.0954 1.2346
1.1033 1.2491
1.1132 1.2678
1.1263 1.2928
1.1445 1.3274
1.1709 1.3764
1.2098 1.4470
1.2681 1.5508
1.3573 1.7082
1.4989 1.9595
1.7395 2.3975
2.2008 3.2817
3.3072 5.6199
7.6349 16.845
98.014 473.89
0.900 0.910
1.8124 1.8961
1.8430 1.9322
1.8803 1.9765
1.9265 2.0320
1.9853 2.1031
2.0615 2.1959
2.1622 2.3192
2.2971 2.4846
2.4796 2.7081
2.7293 3.0139
3.0777 1.4715
3.5805 4.0641
4.3439 5.0191
5.5918 6.6056
7.8640 9.5619
12.741 16.121
26.460 35.515
90.392 133.80
1074.9 1934.4
.18909 + 07 .61138 +
0.825 0.850
1.3217 1.4657
1.3316 1.4801
1.3429 .14968
1.3561 1.5167
1.3718 1.5411
1.3910 1.5715
1.4154 1.6107
1.4472 1.6625
1.4900 1.7325
1.5491 .18286
1.6319 1.9626
1.7499 2.1532
1.9226 2.4332
2.1858 2.8661
2.6140 3.5905
3.3879 4.9611
5.0517 8.1347
9.9695 18.789
39.665 102.33
2612.0 17294.
0.920 0.930
1.9871 2.0871
2.0300 2.1386
2.0831 2.2031
2.1503 2.2856
2.2372 2.3933
2.3515 2.5360
2.5039 2.7269
2.7084 2.9828
2.9845 3.3284
3.3628 3.8026
3.8947 1.9626
4.6743 5.4677
5.8868 7.0381
7.9390 9.7538
11.863 15.103
20.898 27.942
49.139 70.675
206.26 334.82
3699.6 7646.3
.22329 + 08 .95242 + 08

600
TABLES OF SYMMETRIC q-STABLE FRACTILES
0.940
0.950
0.960
0.970
0.975
a = 2.0
Q= 1.9
a = 1.8
a= 1.7
a = 1.6
a = 1.5
a = 1.4
Q= 1.3
a= 1.2
Q= 1.1
a = 1.0
a = 0.9
a = 0.8
a = 0.7
a = 0.6
a = 0.5
a = 0.4
a = 0.3
a = 0.2
a = 0.1
2.1988
2.2616
2.3411
2.4443
2.5804
2.7622
3.0055
3.3314
3.7712
4.3773
3.0777
6.5404
8.6301
12.336
19.889
38.913
106.96
581.72
17500.
.49816 + 09
2.3262
2.4043
2.5049
2.6373
2.8143
3.0519
3.3699
3.7947
4.3687
5.1646
6.3138
8.0679
10.956
16.235
27.440
57.304
173.58
1109.4
46060.
.34462 + 10
2.4758
2.5761
2.7079
2.8850
3.1247
3.4476
3.8778
4.4512
5.2289
6.3181
7.9158
10.408
14.631
22.640
40.504
91.522
311.80
2422.2
.14852 + 06
.35786+11
2.6598
2.7951
2.9785
3.2314
3.5780
4.0427
4.6571
5.4757
6.5953
8.1887
10.579
14.418
21.170
34.607
66.550
166.24
657.79
6553.8
.66067 + 06
.70717 + 12
2.7718
2.9338
3.1584
3.4731
3.9056
4.4814
5.2391
6.2507
7.6445
9.6509
12.706
17.708
26.716
45.198
90.942
241.92
1051.7
12253.
.16884 + 07
.46157 + 13
/
0.980
0.985
0.990
0.995
0.998
a = 2.0
a = 1.9
a= 1.8
a= 1.7
a = 1.6
a = 1.5
a= 1.4
a= 1.3
a = 1.2
a = 1.1
a = 1.0
a = 0.9
a = 0.8
a = 0.7
a = 0.6
a = 0.5
a = 0.4
a = 0.3
a = 0.2
a = 0.1
2.9044
3.1049
3.3915
3.8003
4.3601
5.0967
6.0628
7.3590
9.1641
11.801
15.895
22.754
35.476
62.569
133.01
381.98
1861.9
26241.
52907 + 07
.45292 + 14
3.0690
3.3309
3.7221
4.2893
5.0524
6.0421
7.3404
9.0997
11.589
15.300
21.205
31.409
51.060
94.972
216.61
686.14
3872.9
69680.
.22887 + 08
.84711 + 15
3.2900
3.6691
4.2768
5.1519
6.2841
7.7364
9.6588
12.313
16.160
22.071
31.821
49.411
85.139
170.56
429.22
1559.7
10813.
.27395 + 06
.17838+09
.51424+ 17
3.6428
4.3676
5.6428
7.2896
9.3323
11.983
15.595
20.774
28.630
41.348
63.657
106.99
203.38
461.94
1373.6
6302.5
61965.
.28095 + 07
.58571 + 10
.55411 +20
4.0703
5.9412
8.7634
12.030
16.158
21.735
29.707
41.773
61.216
94.961
159.15
296.57
640.98
1716.5
6355.9
39630.
.61710
.60197 + 08
.58083 + 12
.54473 + 24
-06

TABLES OF SYMMETRIC a-STABLE FRACTILES
601
Table of extreme upper fractiles of the SaS law
.9990 .9991 .9992 .9993 .9994
a = 2.0
a= 1.9
a= 1.8
a= 1.7
a= 1.6
a= 1.5
a = 1.4
a = 1.3
a= 1.2
a= 1.1
Q = 1
a = 0.9
a = 0.8
a = 0.7
a = 0.6
a = 0.5
4.3702
8.084
12.59
17.85
24.72
34.32
48.57
71.04
108.9
178.2
318.3
640.9
1526.
4626.
.2021 + 05
.1588 + 06
4.4143
8.498
13.32
18.97
26.38
36.80
52.35
77.02
118.9
196.1
353.6
720.5
1741.
5378.
.2409 + 05
.1961+06
4.4631
8.993
14.18
20.30
28.37
39.78
56.92
84.30
131.2
218.3
397.9
821.3
2017.
6365.
.2933 + 05
.2483 + 06
4.5179
9.596
15.24
21.93
30,81
43.46
62.60
93.41
146.6
246.4
454.7
952.7
2384.
7703.
.3664 + 05
.3244 + 06
4.5805
10.35
16.57
23.99
33.91
48.14
69.87
105.1
166.7
283.5
530.5
1131.
2890.
9602.
.4738 + 05
.4416 + 06
.9995 .9996 .9997 .9998 .9999
a = 2.0
a= 1.9
a= 1.8
Q= 1.7
a = 1.6
Q= 1.5
Q= 1.4
Q= 1.3
Q= 1.2
a = 1.1
a= 1
a = 0.9
a = 0.8
a = 0.7
a = 0.6
a = 0.5
4.6535
11.33
18.29
26.67
37.97
54.34
79.56
121.0
194.0
334.6
636.6
1385.
3631.
.1246 + 05
.6422 + 05
.6360 + 06
4.7416
12.68
20.66
30.37
43.61
63.02
93.27
143.6
233.6
409.8
795.8
1774.
4799.
.1714 + 05
.9316 + 05
.9939 + 06
4.8530
14.67
24.18
35.92
52.16
76.30
114.5
179.1
296.9
532.3
1061.
2443.
6876.
.2585 + 05
.1505 + 06
.1767 + 07
5.0064
18.06
30.22
45.53
67.15
99.93
152.9
244.6
416.1
769.4
1591.
3833.
11412.
.4612 + 05
.2955 + 06
.3964 + 07
5.2595
25.86
44.28
68.31
103.4
158.4
250.4
416.0
739.7
1441.
3171.
8236.
26943.
.1228 + 06
.9392 + 06
.1590 + 08

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Subject index
a-stable LeVy motion, see self-similar
process
a-stable process
definition, 112
ergodic properties, 580-582
extrema, see probability tails of the
extrema
Markov, 138, 139,578
stationary, see stationary SaS
process
subordinated to another process,
363, 553,556
suprema, see probability tails of the
extrema
zero-one laws, 436-439, 502, 592
a-stable random measure, 118-121
complex-valued, see symmetric a-
stable random measure
control measure of, 119
series representation, 145-148
skewness intensity of, 119
stochastic integral with respect to,
113-118,121-135,149-152,
155-167
symmetric, see symmetric a-stable
random measure
totally skewed to the right, 501
a-stable random variable, 2-10
characteristic exponent, see index
of stability
characteristic function of, 5
different parametrizations, 7, 8
cumulative distribution function,
35-41
density, 9, 35^1, 49, 573-574
unimodality, 574
domain of attraction, 5, 50, 574,
575, 589
domain of normal attraction, 5,24,
34, 552, 573, 585
equivalent definitions of, 2, 3, 5
heavy tails, 16
index of stability, 3
scale parameter, 1
shift parameter, 1
skewness parameter, 1
strict, 3, 12, 13
subordinator, 13, 29, 51, 54
support of, 13, 16, 51
symmetric, see symmetric a-stable
random variable
totally skewed to the left, 13, 241
totally skewed to the right, 13, 15,
20,51,77,241,245
a-stable random vector
associated, 204-208,220,221,579
definition of, 57
density, 74, 79, 81,109,270
existence of conditional moments,
224-236, 270, 582-585
integral representation, 131-135
non-linear regression function,
236-251
graphs, 255-259
spectral measure, 66-68, 71, 85,
107,116
discrete, 70, 107
with respect to different norms,
71
spectral representation, 66
strict, 57-59,72
symmetric, see symmetric a-stable
random vector, 57-59,72
with independent components, 68

622 SUBJECT INDEX
a-stable stochastic integral, see
stochastic integral
defined as a stochastic process, 114
absolute continuity, 524-533, 594
definition of, 524
of a harmonizable process, 532
of a linear fractional stable motion,
533
of a moving average, 532
ARIMA, 380
ARMA, 376, 378, 379, 589, 590
codifference of, 380, 589
covariation of, 380
invertibility, 379
associated measure, see Chentsov
random field
association, 204-208, 220, 221, 579
negative, 207-208
autocovariance function, 318
autoregressive time series, see ARMA
base, see topology
Belyaev's alternative, 460,486
boundedness, 427, 428, 435, 438^143,
447^55,460-470,476,483,
486,491,494,556,593
a counterexample, 461
of a linear fractional stable motion,
446, 452, 494, 545
of a self-similar process, 486,550-
556, 594
of a Takenaka process, 453, 552
via metric entropy, 538-550, 594
Brownian motion, see self-similar
process
Brownian sheet, 394
Cauchy distribution, 10, 41, 49, 52,493
multivariate, 56, 79, 81, 109
change of variables formula, 130
characteristic function of a stable
distribution
multivariate, 65
defined by an a-stable
stochastic integral, 114
univariate, 5
defined by an a-stable
stochastic integral, 117
Chentsov random field, 394-399, 417,
590, 591
associated measure, 394
properties of, 405-407
self-similar, 407-409, 417
Chentsov-type random field, see
Takenaka random field
circular control measure of a complex
stable random measure, 272-
274
codifference, 103-106, 110, 418, 576
definition of, 103
of a linear fractional stable noise,
368
of a log-fractional stable noise, 369
of a moving average, 211-213
of a stationary real harmonizable
process, 305-306
of a stationary SaS process, 208-
215
of a stationary sub-Gaussian
process, 214
of a stationary sub-stable process,
215
ofARMA,380,589
of fractional ARIMA, 384, 590
of linear fractional stable noise,
588
of Ornstein-Uhlenbeck process,
209, 579
of Takenaka random field, 410-
414,591
relation to mixing properties, 213,
580
complex harmonizable fractional stable
motion, see self-similar
process
complex-valued, see symmetric a-stable
random variable or measure
computer program
for computing a-stable probability
density functions, 255
for computing bivariate regress-
sion, 255, 585
for generating a-stable random
variables, 46

SUBJECT INDEX
623
Condition S, 153, 154, 169, 431, 569,
570
conditional normality
absence of, 460
of a real stationary harmonizable
process, 302, 585
of a symmetric a-stable process,
578
of a symmetric a-stable processes,
152-154
of a symmetric a-stable random
variable, 21
continuity, 427-429, 435, 438-443,
455-470,481,483,486,491,
494, 593
of a fractional Brownian motion,
490
of a harmonizable process, 468
of a linear fractional stable motion,
446, 494, 545
of a self-similar process, 486, 550-
556, 594
pointwise, see pointwise continuity
via metric entropy, 538-550, 594
control measure
of a complex-valued symmetric a-
stable random measure, 274
of an a-stable random measure,
119
covariation, 87-95, 110, 127, 174-181,
576
definition of, 87, 88
of a real harmonizable process, 255
of a sub-Gaussian process, 89, 253
ofARIMA,380
of fractional ARIMA, 385
covariation norm, 95-97, 110, 168, 576
definition of, 95
covering number, 538, 556
critical phenomena, see renormalization
group transformation
difference operator, 380
different stochastic processes, meaning
of, 317
domain of attraction, see moving
average-convergence of
partial sums, 589
of a process, 587
of an a-stable random variable, see
a-stable random variable
ergodic properties, see symmetric a-
stable random variable
estimation
of the parameters of a stable
distribution, 40,574
of the parameters of a moving
average and an ARMA, 589
existence of conditional moments
of an a-stable random vector, 224—
236, 582-585
extrema, see probability tails of the ex-
trema
fractiles, see numerical tables
fractional ARIMA, 380-385, 387, 590
codifference of, 384, 590
covariation of, 385
invertibility, 384
fractional Brownian motion, see self-
similar process
continuity, 490, 552
fractional Gaussian noise, 332-339,388,
587
simulation, 370, 588
fractional stable noise, 366-370
linear fractional stable noise, 366,
413
codifference of, 368
log-fractional stable noise, 367
codifference of, 369
long-range dependence, 382
real harmonizable fractional stable
noise, 367,413
sub-Gaussian
fractional stable noise, 367,
413
generalized Chentsov random field, see
Chentsov random field
goodness-of-fit tests, 589
harmonizable fractional stable motion,
see self-similar process
harmonizable process

624 SUBJECT INDEX
absolute continuity, 532
complex-valued, 285, 291-300
conditions for stationarity, 292
continuity, 468, 550
inversion formula for, 519
real, 141,281,300-304,306
codifference of, 305-306
conditional normality of, 302,
585
conditions for stationarity, 301
covariation of, 255
existence of conditional
moments, 254
regression, 254
representation of fractional
Brownian motion, 325-332
representation of fractional
Gaussian noise, 332
Hermite process, see self-similar process
independently scattered, 118, 119, 155,
272, 306, 325, 577
definition of, 119
index of stability, 3, 59
indistinguishable, see version
infinitely divisible, 65, 66, 471-473,
573,574,579,581,591,593,
595
integrability of the sample paths of an
Q-stable process, 502-515
zero-one law, 502
integral representation
of a finite-dimensional Q-stable
vector, 131-135
of a sub-Gaussian process, 142
of a sub-stable process, 144
of an a-stable process, 568-570
on a countable set, 560-568
strongly separable, 459
invertibility
ofARMA, 379
of fractional ARIMA, 384
isotropic, see symmetric Q-stable
random variable or measure
James orthogonality, 97-103, 180, 576
Joseph effect, 309, 311
Khinchine inequality, 508, 518, 533
Ky Fan distance, 431,432, 434, 442
LeVy Brownian motion, see self-similar
random field
Levy distribution, 10,41, 50, 52, 109
LeVy fractional Brownian field, see self-
similar random field
LeVy stable motion, see a-stable LeVy
motion
LeVy-Chentsov random field, see self-
similar random field
LeVy-Khinchine representation, 6, 15,
52
Laplace transform, 14, 15, 20, 78, 100,
143, 186,252
LePage representation, see series
representation
level crossing, 582
by a stationary sub-Gaussian
process, 215
linear fractional LeVy motion, see linear
fractional stable motion
linear fractional stable motion, see self-
similar process
linear fractional stable noise, 366,413
codifference of, 368, 588
simulation, 370, 588
linear regression for a symmetric a-
stable random vector, 174,
578
local continuity, see pointwise continuity
log-fractional LeVy motion, see log-
fractional stable motion
log-fractional stable motion, see self-
similar process
log-fractional stable noise, 367
codifference of, 369
log-stable, 52
long memory, see long-range
dependence
long-range dependence, 335, 336, 339,
345, 366, 376, 382,575,587,
590
measurability, 430-^34, 443, 498-502,
593
definition of, 430

SUBJECT INDEX
625
role of the two-dimensional
distributions, 431
measurable modification, 498
metric entropy, 538-550
definition of, 538
mixing properties, see ergodic properties
moments
conditional, see existence of
conditional moments
existence of, 18
joint, 200
moving average
absolute continuity of, 532
boundedness and continuity of,
468, 593
codifference of, 211-213
continuous time, 138, 214, 215,
221, 306
existence of conditional
moments, 251
generalized, 591
convergence of partial sums, 575,
587
ergodicity, 580
existence of measurable version,
502
Gaussian, with long-range
dependence, 336, 587
representation of fractional Brow-
nian motion, 320-325, 587
representation of fractional
Gaussian noise, 332
stable,
with long-range dependence,
see fractional ARIMA, 590
time series, see ARMA
multifractals, 52
negative dependence, 336, 345, 370
Noah effect, 2, 310
numerical tables, 35, 39, 573, 597-601
Omstein-Uhlenbeck process, 138, 578
codifference of, 209,579
regression, 252
reverse, 139
oscillation, 476
function, 481,493-496
level sets of, 484
non-random, 479
on a set, 478
process, 476,593
pairwise independence, 129,135, 352
pointwise continuity, 430,439-441,443,
459,470,476,481,483
Poisson process, 21,420
different versions, 420
Poisson random measure, 155, 170
Poisson representation of a stochastic
integral, 155-167
probability tails
definition, 16
in the series representation, 188
of a Gaussian distribution, 16
of an a-stable distribution, 16
of the Lp-norm distribution, 515-
519
of theextrema, 187-199,470-475,
579, 593
Rademacher random variable, 287, 456,
500, 506, 528, 533, 555
defined, 23
random field, see self-similar random
field
random measure, see a-stable,
symmetric, Poisson
used in the harmonizable
representation of FBM, 326
Rayleigh distribution, 289
real harmonizable fractional stable
noise, 367,413
real harmonizable process, see
harmonizable process
regression
asymptotic relations for, 246
for a sub-Gaussian random vector,
252
for an Omstein-Uhlenbeck random
vector, 252
for harmonizable vectors, 254
formulas, 260
linear, see symmetric a-stable
random vector

626 SUBJECT INDEX
multiple, see symmetric a-stable
random vector
non-linear, 236-251
graphs, 255-259
regularly varying, 188, 590
relationship between families of a-stable
processes
real harmonizable processes and
moving averages, 306
real harmonizable processes and
sub-Gaussian processes, 302
relative compactness
in the topology of weak
convergence, 561
of a set, 460
renormalization group transformation, 4,
309, 339, 587
rotationally invariant, see symmetric
a-stable random variable or
measure
sample boundedness, see boundedness
sample continuity, see continuity
scale parameter, 1
alemate notation, 97
self-similar process, 311-317, 387-390,
494, 578, 586-592
a-stable L6vy motion, 113, 135,
251,312,349-352,388,587
insight into, 151
integrability of sample paths,
510
sample paths, 440, 552
boundedness and continuity, 486,
550-556, 594
Brownian motion, 311, 313, 314,
320, 387
harmonizable representation of,
329
integrability of sample paths,
509
complex fractional stable motion,
588
complex harmonizable fractional
stable motion, 358-363
definition of, 311
fractional Brownian motion, 140,
318-339,387,388,578,586
harmonizable representation of,
325-332
moving average representation
of, 320-325
simulation, 370
well-balanced representation,
325
Hermite process, 366
Kesten-Spitzer process, 552
linear fractional stable motion,
140, 168,343-349,358,413,
578, 587, 594
absolute continuity, 533
boundedness and continuity,
446, 545, 552
integrability of sample paths,
510
long-range dependence, 345
simulation, 370, 588
unboundedness of sample paths
when H < l/a,452, 552
well-balanced, 140, 344, 552
log-fractional stable motion, 141,
168, 352-355, 388, 588
unboundedness of sample paths,
453, 546, 552
properties of, 340-343
real harmonizable fractional stable
motion, 355-358,413,588
real harmonizable stable motion,
388
self-affine, see with stationary
increments
sub-Gaussian fractional stable
motion, 413
subordinated to another self-
similar process, 364-366,
552, 553, 556
Takenaka random field, 417, 418
with stationary increments, 314
self-similar random field, 392-393, 591
Chentsov random field, 396, 407-
409, 417
definition of, 392
L6vy Brownian motion, 402, 590
L6vy fractional Brownian field,
393, 396,405,590

SUBJECT INDEX
627
Levy-Chentsov random field, 400-
402
Takenaka random field, 405
with stationary increments, 392
semistable law, 585
separability in probability, see Condition
S
separability of a metric space, 427
separability of a stochastic process, 421-
426
definition of, 421
strong, 422
separable kernel, 453
stongly, 459
separant, 421
strong, 422
series representation
of a-stable integrals, 149-152
ofa-stable random measures, 145-
148
of an a-stable random variable,
21-35
of complex-valued stable random
measures and integrals, 286-
291
signed power, definition of, 87
simulation of fractional motions and
noises, 370, 588
simulation of stable random variables,
41
skewness intensity of an a-stable
random measure, 119
Slepian inequalities, 211, 579
slowly varying, 5, 50, 51, 589
spectral measure of an a-stable random
vector, 67, 68, 71, 85, 107,
116
definition of, 66
discrete, 70, 107
Gaussian vector, 76
sub-Gaussian vector, 79, 82
with respect to different norms, 71
stable, see a-stable
stationary increments, 314
in the strong sense, 392
stationary SaS process, 208, 291, 460,
486, 550, 580, 582, 585
codifference of, 208-215
harmonizable, 292, 300, 305, 532,
550
moving average, 212, 221, 306,
532
Omstein-Uhlenbeck, 209
sub-Gaussian, 213, 215, 302
stochastic integral
with respect to a complex-valued
a-stable random measure,
275-281, 307
with respect to an a-stable random
measure, 113-118, 121-135,
149-152, 155-167
as a Poisson integral, 155-167
with respect to an isotropic a-
stable random measure, 281-
286
stochastic ordering, 16, 478, 555
sub-Gaussian fractional stable noise,
367,413
sub-Gaussian process, 142-143, 213,
215,218,302,502,578,580,
582,585
codifference of, 214
covariation of, 89, 253
existence of conditional moments,
252, 585
integral representation, 142
regression, 252
series representation, 169
sub-Gaussian random vector, 78-84, 86,
91, 99, 102, 180, 252, 576,
579
in limit of normalized peri-
odogram, 589
spectral measure, 79, 82
sub-stable process, 143-145, 502, 578
codifference of, 215
integral representation, 143
sub-stable random vector, 168
subordinator, see a-stable random
variable
suprema, see probability tails of the
externa
symmetric a-stable process
conditional normality, 152-154,
578
not conditionally Gaussian, 460

628 SUBJECT INDEX
symmetric a-stable random measure,
121
complex-valued, 272-275, 306,
307
circular control measure of,
272-274
control measure of, 274
integral with respect to, 275-
286, 307
isotropic, 274, 292, 307
rotationally invariant, see
isotropic
symmetric a-stable random variable, 20
characteristic function of, 20
complex-valued, 84-87
isotropic, 84, 109
rotationally invariant, see
isotropic
conditional normality, 21
symmetric a-stable random vector
conditional laws, 181-187,579
linearity of regression, 174, 578
multiple regression, 176-181, 579
tables, see numerical tables
tail behavior, see probability tails
Takenaka random field, 402-405, 414-
417,590,592
codifference, 410-414
codifference of, 591
self-similar, 405,417, 418
unboundedness of sample paths,
453, 552
tight sequence, 61, 107, 561, 562
topology, 422
totally skewed to the left, see a-stable
random variable
totally skewed to the right, see a-stable
random variable
type G, 581
unboundedness, see boundedness
examples, 452
underlying vector, 78
upper-tail probabilities, see numerical
tables
vague convergence, 199
version, 420-421
indistinguishable, 421
measurable, 431, 443
Poisson process, 420
separable, 423,439, 440
weak convergence, 61, 552
zero-one laws, 434, 502, 592
Zolotarev integrals for an a-stable c.d.f.,
39

Author index
Adenstedt, R., 590
Adler, R. J., 211, 578, 579, 582, 589
Ahlfors,L.V.,51
Alarn, K., 207
An, H. Z., 589
Araujo, A., 515
Astrauskas, A., 368-370, 575, 576,587,
588
Avram, R, 575, 590
Bachelier, L., 586
Balkema, A. A., 593
Becker, R. A., 574
Belyaev.K. Y.,460,486
Beran, J., 587
Bhansali, R. J., 589
Billingsley, P., 61,113
Box.G. E. P., 42
Breiman.L., 139,441,572
Bretagnolle, J., 570, 576-578, 595
Brindley, J. E. C, 207
Brockwell, P. J., 215, 389, 589, 590
Brothers, K. M., 39, 573, 597
Brown, G. W., 41
Byczkowski, T., 576
Cambanis, S., 102, 576, 578-580, 583-
588,591,593,594
Cartier, P., 576, 590
Cauchy,A.,571
Chambers, J. M., 8, 26, 42,43, 46, 574
Chen.R., 108
Chen, Y., 574
Chen, Z. G., 589
Chentsov,N.N.,391,590
Chemin, K. E., 574
Choquet-Bruhat, Y., 387
Christoph, G., 572
Cioczek-Georges, R., 230, 231, 583,
585,591
Cline, D. B. H., 589
Cohn, D. L., 432
Cox, D. R., 587
Crame>,H.,216, 582
Crow, E. L., 39, 573
Dabrowski, A. R., 579
Dacunha-Castelle, D., 570, 576-578,
595
Dahlhaus, R., 587
Davis, R. A., 215, 389, 575, 589, 590
Davydov, Y. A., 587
Delehanty, T. A., 574, 597
Desbiens, J., 576
de Acosta, A., 473, 575, 592
de Haan, L., 593
Dobrushin, R. L., 587
Doob, J. L., 423, 424, 592
Dudley, R. M., 538, 539, 575, 592, 594
Dunford,N.,513
Durrett.R., 571,572
DuMouchel, W. H., 8, 35, 39, 42, 573,
574, 597
Esary, J. D., 579
Evertsz, J. G., 52
Fakhre-Zakeri, I., 585
Fama, E., 574
Feldheim, E., 575
Feller, W., 1-3, 5, 6, 14, 16, 17, 24, 25,
34,162,308,338,493,572
Ferguson, T., 573
Femique, X., 539,594
Fix, E., 578
Fox, R., 587

630
AUTHOR INDEX
Gadrich, T., 582, 589
Gangolli, R., 590
Gine\E.,65,515
Gnedenko, B. V., 1, 2, 5, 6, 572, 574
Gradshteyn, I. S., 329
Granger, C. W. J., 588, 590
Gross, A., 213,581
Gross, S., 589
Gupta, V. K., 15
Hahn, M. G., 65
Hall, P., 7, 572, 574
Hannan, E. J., 589
Hardin, Jr., C. D., 18,21,102, 224, 255,
573,576-581,583-585,595
Hill, B. M., 574
Hipel, K. W., 589
Hoffmann-J0rgensen, J., 431
Holcomb, E. W., 574
Holt, D. R., 39, 573
Holtsmark, J., 573
Hosking, J. R. M., 590
Hunt, G, A., 586
Hurst, H. E., 586
Ibragimov, I. A., 246, 574
Ito, K., 476, 487, 494, 495, 507, 577,
578, 593
Jain, N. C, 487, 496, 593
Jakubowski, A., 575, 579
James, R. C, 97
Janicki, A., 572
Jesiak, B., 572
Joag-dev, K., 579
Joyeux, R., 590
Kono, N., 588, 593-595
Kallenberg, O., 513, 577
Kallianpur, G., 487,496, 593
Kanter, M., 346-348, 574-576, 578,
579, 589, 592
Kasahara, Y., 575,588
Kawata, T., 227
Kelley, J. L., 429
Kesten, H., 552, 553
Khinchine, A., 2, 572, 573
Klass, M, 573
Kluppelberg, 589
Knight, K., 589
Kokoszka, P. S., 370,380,385-387,390,
580,581,589-591
Kolmogorov, A. N., 1, 2, 5, 6, 572, 574,
578, 586, 587
Koutrouvelis, L. A., 574
Krakowiak, W., 577
Krivine, J. L., 570, 576-578, 595
Kuelbs, J., 66, 575, 595
Kwapiert, S., 577
Laha, R. G., 66, 572
Lamperti, J. W., 309, 312, 586
Leadbetter, M. R., 215, 216, 582
Ledoux, M, 211, 540, 594, 595
Lee, M.-L. T., 579
Leitch, R. A., 574
Leland.W. E., 587
LePage, R., 573, 583, 585
Levy, J. B., 368-370, 579, 588
Linde, W., 429, 554, 578, 579, 593
Lindgren.G., 216,582
Linnik, Y. V., 246
Liu, J., 589
Loretan, M., 574
Lovejoy, S., 52
Lukacs, E., 52, 247, 572
Levy, P., 2, 3, 112, 391, 571-573, 575,
576, 585, 590
Maejima, M, 575, 576, 578, 586-588,
590, 593-595
Major, P., 320, 339, 587
Mallows, C, 8,42, 43, 46, 574
Mandelbrot, B. B., 2, 52, 309, 311, 312,
573, 578, 586-588, 590
Mandrekar, V., 591, 593
Marcus, D. J., 63, 550, 575, 578, 585,
593, 594
Marques, M., 576
Maruyama, G., 213, 580, 581
Mathe\ P., 579
McConnell, T. R., 577, 594
McCulloch, J. H., 574
McKean Jr., H. P., 590
McLeod, A. L, 589
Mijnheer, J. L., 50

AUTHOR INDEX
631
Mikosch, T., 589
Miller, G., 176, 218, 576, 579, 593, 594
Mittnik, S., 574
Mori, T., 590
Muller, M. E., 42
Nisio, N., 476,487,494, 495, 507, 593
Nolan, J. P., 46, 550, 574, 576, 593, 594
O'Brien, G. L., 586
Paulauskas, V. J., 576
Paulson, A. S., 39,573,574,597
Perlman, M. D., 579
Pisier, G., 550, 578, 585, 594
Pitt, L. D., 207,579
Podg6rski, K., 580, 582
Press, S. J., 79
Proschan, R, 579
P61ya,G.,571
Rachev, S. T., 574, 579
Rajput, B., 576
Ramachandran, B., 227, 582, 583
Rao, C. R., 227,583
Reed, M., 472
Resnick,S.I.,471,575,589
Rice, S.O., 216, 582
Robinson, P. M., 587
Rohatgi, V. K., 66, 572
Roll, R., 574
Rootz6n,H.,215,216, 582
Rosenblatt, M., 587
Rosinski, J., 471, 506, 516, 573, 577,
581,591,593,594
Ross, S. M., 21
Rossberg, H. J., 572
Royden, H. L., 560, 566
Rozanov.Y. A., 581
Ryzhik, I. M., 329
Samarov, A., 587
Samorodnitsky, G., 210, 224, 231,
255,388,516,575,577-579,
582-585,587,588,590,593,
594
Samotij, K., 575
Sato, Y., 415,417,418,591
Saxena, K. M. L., 207
Schertzer, D., 52
Schilder, M., 577, 578, 595
Schoenberg, I. J., 576
Schreiber, M., 570, 595
Schwartz, J. T., 513
Schwartz, L., 554
Shepp, L. A., 108, 509
Sherman, R., 587
Siegel, G., 572
Simon, B., 472
Sinai, Y. G., 587
Soltani, A. R., 586
Spitzer,F.,552,553,571
Steiger, W. L., 589
Stuck, B. W., 8, 42,43, 46, 574
Surgailis, D., 577, 591
Takashima, K., 595
Takenaka,S.,590,591
Talagrand, M., 211, 540, 547, 550, 594,
595
Taqqu, M. S., 210, 224, 230, 231, 255,
309, 320, 336, 366, 368-
370,380,385-388,390,575,
577,578,581,583-585,587,
589-591,594
Thompson, J. W. A., 207
Titchmarsh, E. C, 246,247
Tortrat, A., 579
Tukey, J. W., 41
Van Ness, J. W., 578,586, 587
Varadhan, S. R. S„ 66
Vervaat, W., 387, 573, 586, 588, 595
Walkup, D. W., 579
Wallis, J. R., 309, 586, 588
Waymire, E., 15
Welsh, A. H., 574
Weron, A., 102,572,573,576,580,581,
593, 594
Wiener, N., 577
Wilks, A., 574
Willinger, W., 587
Wilson, D. V., 587
Wintner, A., 574
Wolf, W., 572

632
AUTHOR INDEX
Wolpert, R., 578, 587 Yamazato, M, 574
Woyczynski, N. A., 577
Wu, W., 583,584 Zak, T., 575
Zolotarev, V. M., 2, 4, 7, 9, 18, 39, 40,
Yaglom, A. M, 586, 587, 590 572, 573, 597
Yajima, Y., 587 Zygmund, A., 587