CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 68 EDITORIAL BOARD W. FULTON, T. TOM DIECK, P. WALTERS

LEVY PROCESSES AND INFINITELY DIVISIBLE DISTRIBUTIONS

Already published 1 W.M.L. Holcombe Algebracic automata theory 2 K. Peterson Ergodic theory 3 P.T. Johnstone Stone spaces 4 W.H. Schikhof Ultrametric calculus 5 J.-P. Kahane Some random series of functions, 2nd edition 6 H. Cohn Introduction to the construction, of class fields 7 J. Lambek & P.J. Scott Introduction to higher-order categorical logic. 8 H. Matsumura Commutative ring theory 9 C.B. Thomas Characteristic classes and the cohomology of finite groups 10 M. Aschbacher Finite group theory 11 J.L. Alperin Local representation theory 12 P. Koosis The logarithmic integral I 13 A. Pietsch Eigenvalues and s-numbers 14 S.J. Patterson An introduction to the theory of the Riemann zeta-function 15 H.J. Baues Algebraic homotopy 16 V.S. Varadarajan Introduction to harmonic analysis on semisimple Lie groups 17 W. Dicks & M. Dunwoody Groups acting on graphs 18 L.J. Corwin & F P Greenleaf Representations of nilpotent Lie Groups and their applications 19 R. Fritsch &. R. Piccinini Cellular structures in topology 20 H. 'Gingen Introductory lectures on Siegal modular forms 21 P. Koosis The logarithmic integral II 22 M.J. Collins Representations and characters of finite groups 24 H. Kunita Stochastic flows and stochastic differential equations 25 P. Wojtaszczyk Banach spaces for analysts 26 J.E. Gilbert & M.A.M. Murray Clifford algrebras and Dirac operators in harmonic analysis 27 A. Frohlich & M.J. Taylor Algebraic number theory 28 K. Goebal & W.A. Kirk Topics in metric fixed point theory 29 J.F. Humphreys Reflection groups and Coxeter groups 30 D.J. Benson Representations and cohomology I 31 D.J. Benson Representations and cohomology II 32 C. Allday & V. Puppe Cohomological methods in transformation groups 33 C. Soule et al Lectures on Arakelov geometry 34 A. Ambrosetti 8t G. Prodi A primer of nonlinear analysis 35 J. Palls & F. Takens Hyperbolicity, stability and chaos at homoclinic bifurcations 36 M. Auslander, I. Reiten & S.O. Smalo Representation theory of Artin algebras 37 Y. Meyer Wavelets and operators I 38 C. Weibel An introduction to homological algebra 39 W. Bruns & J. Herzog Cohen-Macaulay rings 40 V. Snaith Explicit Brauer induction 41 G. Laumon Cohomology of Drinfield modular varieties I 42 E.B. Davies Spectral theory and differential operators 43 J. Diestel, H. Jarchow & A. Tonge Absolutely summing operators 44 P. Mattila Geometry of sets and measures in Euclidean spaces 45 R. Pinsky Positive harmonic functions and diffusion 46 G. Tenenbaum Introduction to analytic and probabilistic number theory 47 C. Peskin An algebraic introduction to complex projective geometry I 48 Y. Meyer & R Coifman Wavelets and operators II 49 R. Stanley Enumerative combinatorics I 50 I. Porteous Clifford algebras and the classical groups 51 M. Audin Spinning tops 52 V. Jurdjevic Geometric control theory 53 H. Voelklein Groups as Galois groups 54 J. Le Potier Lectures on vector bundles 55 D. Bump Automorphic forms 56 G. Laumon Cohomology of Drinfield modular varieties II 57 D.M. Clark & B.A. Davey Natural dualities for the working algebraist 59 P. Taylor Practical foundations of mathematics 60 M. Brodmann & R. Sharp Local cohomology 61 J.D. Dixon, M.P.F. Du Sautoy, A. Mann & D. Segal Analytic pro-p groups, 2nd edition 62 R. Stanley Enumerative combinatorics II 64 J. Jost & X. Li-Jost Calculus of variations

LEVY PROCESSES AND INFINITELY DIVISIBLE DISTRIBUTIONSKEN-ITI SATO

CAMBRIDGEUNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge, United KingdomCAMBRIDGE UNIVERSITY PRESS

http://www.cup.cam.ac.uk The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA http://ww-w.cup.org 10 Stamford Road, Oaldeigh, Melbourne 3166, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Originally published in Japanese as Kahou Katei by Kinolcuniya, C Kinokuniya 1990. C English edition Cambridge University Press 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in English 1999 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this book is available from the British Library ISBN 0 521 553024 hardback

ContentsPreface Remarks on notation Chapter 1. Basic examples 1. Definition of Lvy processes 2. Characteristic functions 3. Poisson processes 4. Compound Poisson processes 5. Brownian motion 6. Exercises 1 Notes Chapter 2. Characterization and existence of Lvy and additive processes 7. Infinitely divisible distributions and Lvy processes in law 8. Representation of infinitely divisible distributions 9. Additive processes in law 10. Transition functions and the Markov property 11. Existence of Lvy and additive processes 12. Exercises 2 Notes 1 1 7 14 18 22 28 30 31 31 37 47 54 59 66 68

Chapter 3. Stable processes and their extensions 69 13. Selfsimilar and semi-selfsimilar processes and their exponents 69 14. Representations of stable and semi-stable distributions 77 15. Selfdecomposable and semi-selfdecomposable distributions 90 99 16. Selfsimilar and semi-selfsimilar additive processes 17. Another view of selfdecomposable distributions 104 18. Exercises 3 114 Notes 116 Chapter 4. The LvyIt decomposition of sample functions 19. Formulation of the LvyIt decomposition 20. Proof of the LvyIt decomposition 21. Applications to sample function properties 119 119 125 135

vi

CONTENTS

22. Exercises 4 Notes Chapter 5. Distributional properties of Lvy processes 23. Time dependent distributional properties 24. Supports 25. Moments 26. Lvy measures with bounded supports 27. Continuity properties 28. Smoothness 29. Exercises 5 Notes Chapter 6. Subordination and density transformation 30. Subordination of Lvy processes 31. Infinitesimal generators of Lvy processes 32. Subordination of semigroups of operators 33. Density transformation of Lvy processes 34. Exercises 6 Notes Chapter 7. Recurrence and transience 35. Dichotomy of recurrence and transience 36. Laws of large numbers 37. Criteria and examples 38. The symmetric one-dimensional case 39. Exercises 7 Notes Chapter 8. Potential theory for Lvy processes 40. The strong Markov property 41. Potential operators 42. Capacity 43. Hitting probability and regularity of a point 44. Exercises 8 Notes Chapter 9. Wiener-Hopf factorizations 45. Factorization identities 46. Lvy processes without positive jumps 47. Short time behavior 48. Long time behavior 49. Further factorization identities 50. Exercises 9 Notes

142 144 145 145 148 159 168 174 189 193 196 197 197 205 212 217 233 236 237 237 245 250 263 270 272 273 273 281 295 313 328 331 333 333 345 351 363 369 382 383

CONTENTS

vii

Chapter 10. More distributional properties 51 Infinite divisibility on the half line 52. Unimoda1ity and strong unimodality 53. Selfdecomposable processes 54. Unimodality and multimodality in Lvy processes 55. Exercises 10 Notes Solutions to exercises References and author index Subject index

385 385 394 403 416 424 426 427 451 479

PrefaceStochastic processes are mathematical models of random phenomena in time evolution. Lvy processes are stochastic processes whose increments in nonoverlapping time intervals are independent and whose increments are stationary in time. Further we assume a weak continuity called stochastic continuity. They constitute a fundamental class of stochastic processes. Brownian motion, Poisson processes, and stable processes are typical Lvy processes. After Paul Lvy's characterization in the 1930s of all processes in this class, many researches have revealed properties of their distributions and behaviors of their sample functions. However, Lvy processes are rich mathematical objects, still furnishing attractive problems of their own. On the other hand, important classes of stochastic processes are obtained as generalizations of the class of Lvy processes. One of them is the class of Markov processes; another is the class of semimartingales. The study of Lvy processes serves as the foundation for the study of stochastic processes. Dropping the stationarity requirement of increments for Lvy processes, we get the class of additive processes. The distributions of Lvy and additive processes at any time are infinitely divisible, that is, they have the nth roots in the convolution sense for any n. When a time is fixed, the class of Lvy processes is in one-to-one correspondence with the class of infinitely divisible distributions. Additive processes are described by systems of infinitely divisible distributions. This book is intended to provide comprehensive basic knowledge of Lvy processes, additive processes, and infinitely divisible distributions with detailed proofs and, at the same time, to serve as an introduction to stochastic processes. As we deal with the simplest stochastic processes, we do not assume any knowledge of stochastic processes with a continuous parameter. Prerequisites for this book are of the level of the textbook of Billingsley [27] or that of Chung [70]. Making an additional assumption of selfsixnilarity or some extensions of it on Lvy or additive processes, we get certain important processes. Such are stable processes, semi-stable processes, and selfsimilar additive processes. We give them systematic study. Correspondingly, stable, semistable, and selfdecomposable distributions are treated. On the other hand,

PREFACE

the class of Levy processes contains processes quite different from selfsimilar, and intriguing time evolution in distributional properties appears. There are ten chapters in this book. They can be divided into three parts. Chapters 1 and 2 constitute the basic part. Essential examples and a major tool for the analysis are given in Chapter 1. The tool is to consider Fourier transforms of probability measures, called characteristic functions. Then, in Chapter 2, characterization of all infinitely divisible distributions is given. They give description of all Lvy processes and also of all additive processes. Chapters 3, 4, and 5 are the second part. They develop fundamental results on which subsequent chapters rely. Chapter 3 introduces selfsimilarity and other structures. Chapter 4 deals with decomposition of sample functions into jumps and continuous motions. Chapter 5 is on distributional properties. The third part ranges from Chapter 6 to Chapter 10. They are nearly independent of each other and treat major topics on Lvy processes such as subordination and density transformation, recurrence and transience, potential theory, Wiener-Hopf factorizations, and unimodality and multimodality. We do not touch extensions of Lvy processes and infinitely divisible distributions connected with Lie groups, hypergroups, and generalized convolutions. There are many applications of Lvy processes to stochastic integrals, branching processes, and measure-valued processes, but they are not included in this book. Risk theory, queueing theory, and stochastic finance are active fields where Lvy processes often appear. The original version of this book is Kahou katei written in Japanese, published by Kinokuniya at the end of 1990. The book is enlarged and material is rewritten. Many recent advances are included and a new chapter on potential theory is added. Exercises are now given to each chapter and their solutions are at the end of the volume. For many years I have been happy in collaborating with Makoto Yamazato and Toshiro Watanabe. I was encouraged by Takeyuki Hida and Hiroshi Kunita to write the original Japanese book and the present book. Frank Knight and Toshiro Watanabe read through the manuscript and gave me numerous suggestions for correction of errors and improvement of presentation. Kazuyuki Inoue, Mamoru Kanda, Makoto Maejima, Yumiko Sato, Masaaki Tsuchiya, and Makoto Yamazato pointed out many inaccuracies to be eliminated. Part of the book was presented in lectures at the University of Zurich [405] as arranged by Masao Nagasawa. The preparation of this book was made in AMSLaTeX; Shinta Sato assisted me with the computer. My heartfelt thanks go to all of them. Ken-iti Sato Nagoya, 1999

Remarks on notationN, Z, Q, R, and C are, respectively, the collections of all positive integers, all integers, all rational numbers, all real numbers, and all complex numbers. Z+ , Q+ , and R+ are the collections of nonnegative elements of Z, Q, and R, respectively.For r E R, positive means r>0; negative means x < O. For a sequence {x.}, increasing means xn, < x 1 for all n; decreasing means rn > xn-Ei for all n. Similarly, for a real function f, increasing means f (s) < f (t) for s < t, and decreasing means f (s) > f (t) for s < t. When the equality is not allowed, we say strictly increasing or strictly decreasing. Rd is the d-dimensional Euclidean space. Its elements x = y = (0 3= 1,...4 are column vectors with d real components. The inner product is (x,y) = x3 y3 ; the norm is I r = (Ed x j2 )112. The word 3.1 d-variate is used in the same meaning as d-dimensional. For sets A and B, A C B means that all elements of A belong to B. A, BC R d, z E Rd, and c E R, A+ z = {r+ z: x E A}, Az = {x z: A}, A + B = {x + y: x E A, y E B } , A B = {x y: x E A, y E cA = {ex: x E A}, A= {x: x E A}, A\ B = {x: x E A and x and dis(z, A) = infxEA lz xi. 71 is the closure of A. For

x E B}, B},

B(R d) is the Borel a-algebra of Rd . For any B E 13(R d), I3(B) is the (7-algebra of Borel sets included in B. I3(B) is also written as 13B.Leb(B) is the Lebesgue measure of a set B. Leb(dx) is written dr. f g(x,y)c4F(x,y) is the Stieltjes integral with respect to x for fixed y.The symbol kJ represents the probability measure concentrated at a. [p].73 is the restriction of a measure p. to a set B. The expression i 1 *j represents the convolution of finite measures pi and ii2 ; pn = pn* is the n-fold convolution of p. When n =- 0, /Ln is understood to be bo . Sometimes pt(B) is written as pB. Thus p(a,b1 = p((a,b]). A non-zero measure means a measure not identically zero.

1 B (x) is the indicator function of a set B, that is, 1 B (x) = 1 for x E B and 0 for r E B.

xli

REMARKS ON NOTATION

a A b = min-Ca,* a V b = max{a,b}. The expression sgn x represents the sign function; sgn x = 1, 0, 1 according as x> 0, = 0, 0, (t) = limo "(t h) G Rd for t > 0. and (t) has left limits I is the identity matrix. A' is the transpose of a matrix A. For an n x m real matrix A, 11All is the operator norm of A as a linear transformation from Rin to IV, that is HAI = sup 1z1 _1 lAxl. 0,

(1.5)

lim P[IX X t i > el = O.

Stochastic processes are mathematical models of time evolution of random phenomena. So the index t is usually taken for time. Thus we freely use the word time for t. The most basic stochastic process modeled for continuous random motions is the Brownian motion and that for jumping random motions is the Poisson process. These two belong to a class called Lvy processes. Lvy processes are, speaking only of essential points, stochastic processes with stationary independent increments. How important this class is and what rich structures it has will be gradually revealed in this book. First we give its definition. DEFINITION 1.6. A stochastic process {Xt : t > 0} on process if the following conditions are satisfied. is a Lvy

(1) For any choice of n > 1 and 0 < to < t 1 < < tn , random variables Xt o , Xt, XL , X 2 X 1 ,..., Xt Xi_, are independent (independent increments property). (2) Xo = 0 a. s. (3) The distribution of Xs+t X s does not depend on s (temporal homogeneity or stationary increments property). (4) It is stochastically continuous. (5) There is fto E .F with P[9.13] --= 1 such that, for every cv E 9.4), Xt(w) is right-continuous in t > 0 and has left limits in t > O. A Lvy process on lie is called a d-dimensional Lvy process. Dropping the condition (5), we call any process satisfying (1)(4) a Lvy process in law. We define an additive process as a stochastic process satisfying the conditions (1), (2), (4), and (5). An additive process in law is a stochastic process satisfying (1), (2), and (4).

4

1. BASIC EXAMPLES

The conditions (1) and (3) together are expressed as the stationary independent increments property. Under the conditions (2) and (3), the condition (4) can be replaced by

(1.6)

lim P[ IXt l > e]

0

for

e > O.

We will see in Chapter 2 that any Lvy process in law has a modification which is a Lvy process. Similarly any additive process in law has a modification which is an additive process. Thus the condition (5) is not essential. Lvy defined additive processes without assuming the conditions (4) and (5). But such processes are reducible to the additive processes defined above. See Notes at the end of Chapter 2. EXAMPLE 1.7. Let {Xt } be a Lvy process on Rd and h(t) be a strictly increasing continuous function from [0, ce) into [0, co) satisfying h(0) = O. Then IX401 is an additive process on Rd . If h(t) = ct with c > 0, then {Xh(t)} has temporal homogeneity and it is a Lvy process. A theorem of Kolmogorov guarantees the existence of a stochastic process with a given system of finite-dimensional distributions. Let SI -= (Rtz)[o,...), the collection of all functions Le = (ce(t))t[o,.) from [0, ce) into Rd . Define Xt by X(w) = ce(t). A set

(1.7)

C=

X(t i ,ce) E Bi, - ,X(tit,ce) E B n 1

for 0 < t1 < - < tn and B1 ,..., Bn E B(Rd) is called a cylinder set. Consider the rf-algebra generated by the cylinder sets, called the Kolmogorov

cr-algebra.THEOREM 1.8 (Kolmogorov's extension theorem). Suppose that, for any choice of n and 0 < t1 < < tn, a distribution /41,...,4, is given and that, if B1 , . . . , Bn G B(Rd) and Bk = Rd, then

(1.8)

x - - - x Bn)

-=

x---x

Bk_i X Bk+i X ' X Ba)-

Then, there exists a unique probability measure P on .7. that has {At i,...41 as its system of finite-dimensional distributions. This theorem is in Kolmogorov [269]. Proofs are found also in Breiman

[59] and Billingsley [27].Construction of the direct product of probability spaces is often needed. THEOREM 1.9. Let (1l, J, Pn) be probability spaces for n = 1, 2,.... Letil=fli xa2 x--- and let .T be the o--algebra generated by the collection of sets(1.9)

C =- Ice = (wi , ce2,

) : Lek E Ak for k = 1, .. ,n},

1. DEFINITION OF LEVY PROCESSES

5

over all n and all Ak C ..rk for k = 1,... ,n. Then there exists a unique such that probability measure P on

P[C] -=for each C of (1.9).

Pn [An]

Proof is found in Halmos [161] and Fristedt and Gray [134]. If SI = Rd and .F = B(Rd) for each n, then Theorem 1.9 is a special case of Theorem 1.8. We give the definition of a random walk. It is a basic object in probability theory. A Lvy process is a continuous time analogue of a random walk.DEFINITION 1.10. Let {Zn : n = 1, 2, ... } be a sequence of independent and identically distributed R d-valued random variables. Let So = 0, Sn Z for n = 1, 2, .... Then {Sn : n -= 0,1, ... } is a random walk on Rd , 3 .-.1 or a d-dimensional random walk.

En

For any distribution ti on Rd , there exists a random walk such that Zn distribution p. This follows from Theorem 1.9. has Two families {X}, {17,9 } of random variables are said to be independent if, for any choice of t 1 , , 4, and Si, , .5, the two multi-dimensional random variables (X5 ) 3 -1, .,n and ("Ysk )k= 1 ,...,in are independent. A sequence of events {An : n = 1, 2, ... } is said to be independent, if the sequence of random variables {1 A (w): n =- 1, 2, ... } is independent. For a sequence of events {An}, the upper limit event and the lower limit event are defined by lim sup A n = respectively.PROPOSITION 1.11 (Borel-Cantelli lemma). (i) If EnL l P[Anj < co, ' then P[lim supa, An ] = 0. (ii) If {A n : n =- 1, 2, ... } is independent and ati P[A] = c, then we have P[ lim sup, An ] = 1.

n=1 k=n

n

CO CC

Ak

and

lim silif A n = n oo

n=1 k=72.

u n Ak

00

CO

A sequence of Rd-valued random variables {X n : n 1, 2, ... } is said to converge stochastically, or converge in probability, to X if, for each c > 0, P[ IXn XI > E] = 0. This is denoted by Xn X in prob.-

If {Xn } converges stochastically to X and X', then X = X' a. s. A sequence {Xn} is said to converge almost surely to X, denoted by

Xif P[Iim,,,,,Xn(w) =- X(co) ] = 1.

6PROPOSITION

1.

BASIC EXAMPLES

(ii) If X X.

1.12. (i) If X n -4 X a. s., then X n -4 X in prob. X in prob., then a subsequence of {X} converges a. s. to

It follows from (i) that, if {Xt } is a Lvy process, then

(1.10)

X= X_

a. s.

for any fixed t > 0,

where Xt _ denotes the left limit at t. For tn t implies Xt, --> Xt _ a. s. and Xi,, X t in prob. Among the five conditions in the definition of a Lvy process the condition (4) is implied by (2), (3), and (5). In fact, for any tn 0, Xi,, converges to 0 a. s. and hence in prob., which implies (1.6). PROPOSITION 1.13 (Inheritance of independence). Suppose that, for cc. If the family {Xi,n: j = each j = 1, . . . , k, Xi in prob. as n 1, ... ,k} is independent for each n, then the family {X i : j = 1,... ,k} is independent. Proofs of Propositions 1.11-1.13 are found in [27], [70] and others. The concept of independence is extended to a-algebras (though we will not use this extension often). Let (12,T, P) be a probability space. Subu-algebras J , F2 , of T are said to be independent if, for any An E Yn, n = 1,2, ... , {An } is independent. Given a family of random variables {Xt : t E T}, where T is an arbitrary set, we say that a sub-a-algebra g is the a-algebra generated by {Xt : t E T and write g = u(xt : t E T) if (1) Xt is g-measurable for each t, (2) g is the smallest u-algebra that satisfies (1). In general, for a family A of subsets of 5/, the smallest o--algebra that contains A is called the a-algebra generated by A and denoted by a- (A). A random variable X and a-algebra .F1 are said to be independent if a(X) and T1 are independent.... }

THEOREM 1.14 (Kolmogorov's 0-1 law). Let {Tn : n = 1,2, ...} be an independent family of sub-a-algebras of T. If an event A belongs to the a-algebra o-(U,,Tn) for each m, then P[A] is 0 or 1. Proofs are found in [27], [70] and others. The following fact (sometimes called Dynkin's lemma, see [71], [1071) will be used. PROPOSITION 1.15. Let A be a collection of subsets of ft such that

(1) AEAandBEAimplyAnBEA. Let C D A and suppose the following. (2) If A n E C, n = 1,2,..., and {An} is increasing, then U (3) If A E C, B E C, and A D B, then A\ B E C.(4) 11E C. Then C a(A).

1 A,, E C.

2. CHARACTERISTIC FUNCTIONS

7

The proof of the following proposition on evaluation of some expectations shows the strength of Proposition 1.15. PROPOSITION 1.16. Let X and Y be independent random variables on Rdi and 1R('2, respectively. If f (x,y) is a bounded measurable function on Rd' x Rd2 , then g(y) = E[f (X,y)1 is bounded and measurable and E[f(X,Y)] = E[g(Y)]. Proof. Let C be the collection of sets A G ,13(Rd1 x Rd2 ) such that f = 1 A (x y) satisfies the conclusion above. Here ' A is the indicator function of the set A (see Remarks on notation). Let A be the collection of sets A = A1 x if2 with A1 G B(Rd1 ) and A2 G B(Rd2 ). It follows from the definition of independence that A C C. Since A and C satisfy (1)(4) of Proposition 1.15 with 1 = Rd' x Rd2 , we have C = 13(Rdi x R('2 ). For general f use approximation by linear combinations of functions of the form 1 A (x, y).-

2. Characteristic functionsThe primary tool in the analysis of distributions of Lvy processes is characteristic functions, or Fourier transforms, of distributions. We will give definitions, properties, and examples of characteristic functions. DEFINITION 2.1. The characteristic function I (z) of a probability measure ,a on Rd is

z E Rd . Rd The characteristic function of the distribution Px of a random variable X on Rd is denoted by Px (z). That is(2.1)(z)

ii(z) = f

=

f egz,r) px(ds) =Rd

DEFINITION 2.2. A sequence of probability measures converges to a probability measure p, written as

n --- 1, 2, ...

tt

IL

as n

oo,

if, for every bounded continuous function f, f (x)p, z (dx)fRa

f(x),a(dx)

as n 4 co.

When and fi n are bounded measures, the convergence p n A is defined in the same way. When Lai l are probability measures with a real parameter, we say that Ptas

s

t,

8

1. BASIC EXAMPLES

if f (x)ti i (dx) as s t f (x)fi s (dx) Rd fRci for every bounded continuous function f. This is equivalent to saying that ps tit for every sequence sn that tends to t. We say that B is a p-continuity set if the boundary of B has it-measure O. The convergence pn --- y is equivalent to the condition that fi(B) ti(B) for every y-continuity set B E /3(Rd). A sequence of random variables {X n } on Rd converges in probability to X if and only if the distribution of Xn X converges to So (distribution concentrated at 0). The next fact is frequently used. PROPOSITION 2.3. If X n X in probability, then the distribution of X n converges to the distribution of X. DEFINITION 2.4. The convolution of two distributions pi and f2 on Rd , denoted by y pi*Ii2, is a distribution defined by

(2.2)

ti(B) = f f1 B (X

xRd

(dx)y2 (dy),

B E B (Rd) .

The convolution of two finite measures on Rd is defined by the same formula. The convolution operation is commutative and associative. If X1 and are independent random variables on Rd with distributions fi and P2, respectively, then X1 + X2 has distribution pi*P2.X2

The following are the principal properties of characteristic functions. In (v) we will use the following terminology: i is the dual of y and fi the symmetrization (of a probability measure) of y if ii(B) = y(B),si B = {x: z B } , and = [i*ri. When d = 1, another name of the dual of y is the reflection of II. If rt is identical with its dual, it is called

e

symmetric.

PROPOSITION 2.5. Let 1.1, 11 1 , 112 , fin be distributions on Rd. (i)(Bochner's theorem) We have that ii,(0) = 1 and Iii(z)1 5_ 1, and ii(z) is uniformly continuous and nonnegative-definite in the sense that, for each n 1, 2, ... ,n n

(2 .3)3=1 k=1

-zoe.gk

> 0 for

-

E

Rd 67 - en

Conversely, if a complex-valued function cp(z) on Rd with yo(0) = 1 is continuous at z -= 0 and nonnegative-definite, then c,o(z) is the characteristic function of a distribution on R". (ii) If 1 (z) = p2 (z) for z E 1W', then Il i =

2. CHARACTERISTIC FUNCTIONS

9

(iii) If p = fil*/12, then A(z) = , then random variables on

iii(412(z). If X 1 and X2 are independent

Px x2 (z) =(iv) Let X = (Xi)j=1,..., be an d -valued random variable, where X 1 , . . . , X are Rd -valued random variables. Then X 1 ,. , X. are independent if and only if

Px (z) = Px (zi ) ...Px,(z.)

for

z=

zi G

(y) Suppose' that re, is the dual of p and till is the symmetrization of p. = A(z) and I-21 (z) = I(z)I2 . Then ii(z) = (vi) If pi, ) p, then ii.(z) ) pi(z) uniformly on any compact set. (vii) If ii.(z) ) p(z) for every z, then p,, ) p. (viii) If ii.(z) converges to a function c,o(z) for every z and cp(z) is continuous at z = 0, then (p(z) is the characteristic function of some distribution. (ix) Let n be a positive integer. If p has a finite absolute moment of order n, that is, f lxin p(dx) < co, then ii(z) is a function of class C" and, d for any nonnegative integers n1 , . . , nd satisfying n1 + + n < n,

i zd ) x7 1 ...4dp(dx) = [ (-k ) 14 ... ( 1 aand [i(Z)] aJz=0

.

(x) Let n be a positive even integer. If A(z) is of class Cn in a neighborhood of the origin, then p has finite absolute moment of order n. (xi) Let oo < < < oo for j = 1, ,d and B = [ai ,b1 ] x - x [ad, bd]. If B is a A-continuity set, then

p(B) = Jim (27r) -d f[c,cld

ii(z)dz f e- (x'z) dx. 1

(xii) If the

f iii(z)jdz < co, then p is absolutely continuous with respect to Lebesgue measure, has a bounded continuous density g(x), and g(x) = (27r)_ d f ei(z)dz.Rd

The assertion (xi) contains the inversion formula, which strengthens the one-to-one property (ii). In the one-dimensional case the properties above are proved in Billingsley [27], Breiman [59], Chung [70], Fristedt and Gray [134], and many other books. In general dimensions see Dudley [97], pp. 233-240, 255, Cuppens [81], pp. 16, 37, 41, 53, 54, and also Linnik and Ostrovskii [293], pp. 169-173.

10 by (2.4)

1. BASIC EXAMPLES When y is a distribution on [0, co), the Laplace transform of p is defined

L o (u) = f[0,00)

e-"ti(dx)

for u > O.

PROPOSITION 2.6. Let ji,Pi, and 112 be distributions on [0, co). (i) If L i(u) = 1,1,2 (u) for u 0, then pi = /12. (ii) If /1 = p1 *p2 , then L(u) -= L A1 (u)L t,2 (u). Proof (i) For any complex w with Re w < 0 we can define 4 ,i (w) = 1, 2. These are analytic on fw: Re w < 01. For the integral e"pi (dx) is analytic since we can differentiate under the integral sign, f[0,n] and this sequence is uniformly bounded and convergent to 't', (w) pointwise as n > co. If w = u < 0, then 4) 1 (w) = 4) 2 (w). Hence 40 1 (w) = 4)2 (w) on {w: Re w O. This is the case of one-sided strictly stable distributions of index 1/2, which we shall study in Chapter 3. We can check p(F1) -= 1 by

c2/ x =_ y2 as

coc(270 -1/2 f o

e -c2/(2.) x -3/2 dx 2 (270 -1/2

co

e - y2 , 2dy

1.

JoOE9

Let us find its Laplace transformL,(u) c(270 -1 /2 f o

e -ux-c2/(ax) x-3/2 dx.

Differentiation in u > 0 and the change of variables ux = c2/ (2y) lead to co

(u)=- c(2r) -112 f

e-u.-- c21(2.)x -112ds

= c2 (471-u) -1/2 - oe e "-c2/(29) y-3/2dy c(2u)-1/24(u). f

2. CHARACTERISTIC FUNCTIONS

13

Noting that 4(u) is continuous on fu > 01 and L(0) = 1, we see that

(2.11) (2.12)

= exp(c(2u) 1/2 ),

u > 0.

The characteristic function is

iI(z) = exp(-4z1 1/2 (1 isgnz)).

In fact, let 43(w) -= f e"p(dx) for complex w with Re w < 0. As is shown in the proof of Proposition 2.6, (I)(w) is analytic on {Re w 0 is defined by

p(B) = aWe have

f nno,00

edx. -"

(2.13) (2.14)

L o (u) = a/(a + u), ii(z) = c I (cx iz),

u > 0, z E R.

The mean of p is 11a. EXAMPLE 2.15. (d =1) For c> 0 and a >0,

p(B) = (ac Ir(c)) fBn(0,co)

sc-le-"dx

is the r-distribution with parameters c, a. It is exponential if c = 1. We get

(2.15) (2.16)

LI,(u) -= (1 + u > 0, fi(z) = (1 ia-l z) -c = exp[clog(1 ia -l z)],

z E I,

where log is the principal value (that is, the imaginary part is in (ror]). The mean of i is c/a. When c = n/2 with n G N and a = 1/2, statisticians call the x2-distribution with n degrees of freedom.

14

1. BASIC EXAMPLES

EXAMPLE 2.16. (d

= 1) The geometric distribution with parameter p,

0 < p 0,z ER,

= p(1 Ii(z) =p(1 where g = 1 and p, 0 < p

p. The negative binomial distribution with parameters c> 0 0,

z E R.

Notice that the parameter c is not restricted to positive integers. For w = qeiz, (1 w)_c stands for e -c kg(1-w) (log is the principal value). EXAMPLE 2.17. (d = 1) Let n be a positive integer, 0 < p < 1, and g = 1 p. The binomial distribution with parameters n, p is/2{0( 71

0 pk qnk

k

0,1,...,n.

We have = (pe- n + q)n, u 0,

i/(z) = (pe' + q) 1 ,EXAMPLE 2.18. (d is

z E R.

= 1) The uniform distribution on [a, a] for a > 0p(B) = (2a) -1 fBn[- a,a]

dx,

ii(z) = (sin az)/ (az).

with the understanding that (sin az)I (az) = 1 for z = O.EXAMPLE 2.19. The distribution concentrated at a single point y E Rd is the 6 distribution at -y and denoted by 67 . Its characteristic function is-

3. Poisson processes

We define and construct Poisson processes.

3. POISSON PROCESSES

15

DEFINITION 3.1. A stochastic process {X t : t > 0} on IR is a Poisson process with parameter c> 0 if it is a Lvy process and, for t> 0, Xt has Poisson distribution with mean ct.

THEOREM 3.2 (Construction). Let {W n : n = 0, 1, ... } be a random walk on Et, defined on a probability space (St,,F,P), such that Ta = Wn Wn-1 has exponential distribution with mean c> 0. Define X t by (3.1) X(w) =n if and only if W(w) s+tIT> s] = PIT > t],

s > 0, t > 0,

called lack of memory. Here, for an event A with positive probability, P[B IA] is the conditional probability of B given A, that is, (3.3)

P[BIA] =r P[B n AV PO].

The property (3.2) follows easily from the definition of exponential distribution. Conversely, if a nonnegative random variable has the property of lack of memory, then its distribution is either exponential or 60. When we consider a model of arrival of customers at a service station and assume that the length of interval of successive arrivals has lack of memory, WT., is the waiting time until the arrival of the nth customer, Xt is the number of customers who arrived before the time t, and {Xt } is a Poisson process. Proof of Theorem 3.2. The random walk {W} increases to co almost surely, since P[W, < t] < P[Ti < t,...,T < t] = (P[T< tpn * 0, i n-400.

So we can define {Xt} by (3.1). Obviously X0 = 0 a.s. Example 2.15 says that Wn has P-distribution with parameters n, c. We have (3.4) P[Xt = n] = e-a (n!) -1 (ct)?1 , t > 0, n 0. In fact,

P[Xt = n ] = P[W t+siX t =n]= e's, Calculation of the same sort as used for (3.4) leads to (3.5) P[X = n, Wn+1 > t + .9 ] = P[Wn < t, Wn Tn+i > t Si= Cn+1 ((n 1)!) -1

f

et-le

-CXdX

e -Cydyt+s--.

e-C(t+.9) (n!)

-1 (con .

This and (3.4) yield (3.5) by the definition of the conditional probability. Let n > 0 and m > 1. Let us consider the conditional distribution of (Wn.+1 t7 Tn+2, .. , Tn+m) given Xt = n. It is equal to the distribution of T2 ,... , Tm). To show this, let P[Wn < t < Wn+1] = a and observe that, for any Si,.., 5m >

P[ Wn+1 t > Sl, T.+2 > S2, - , T.+7. > s, IXt =n]

= P[ wn < t,,

Wn+1 t > 3 17 Tn+2 > $2, ,Tn+m > Sm1I a

= P[W < t Wn+i t> Si]P[Tn+2 > s2, , Tn+77, > Sm I/a n = P[Wn+i t > s i I Xt = n]P[T > s2,... ,Tn+ > s ] it+2 yn

= P[T > si]P[T2 > sz, i > stn ] = P[Ti > Si, T2> $2, , T. > sn .]

Here we have used (3.5). Now it follows that

(3.6)In fact,

P[Xt+s X t =

= P[X, = m],

t > 0, s > O.

P[Xt = n, Xt+s X t = m] = P[X = n, Xt+, = n + m] t = P[Xt = n]P[147 arn < t S < Wn+771+1 I Xt = n] = P[Xt = n]P[W, s < Wm+11= P[Xt = n1P[X. =where we have written Wn+ni O. The conditional distribution of W1, - - - ,W. given that X t = n coincides with the distribution of the order statistics V1 < V2 < 0 a. s. in t > 0 by the definition of Y(t). We claim that Y(t) Define

s-1=

n=1 tn=1 tEQC1(0,1/m)

n u n {, x (CO CO

t) ,

K]= P[X(1)> t -112 K] > 1/2, nfor any K. By Fatou's lemma

n > co,

P[ X(4) > K for infinitely many n]

5. BROWNIAN MOTION

25

=Hence P[ lim

E[

lim sup 1 { x (t) >ns oo

lim sup E[1{x() >K1] = 1/2.ns oo

X (t) > K] > 1/2. Therefore P[ limsup X(t n) =- co} > 1/2.tts co

Let to = 0 and let Z, = X(t) X(4,1). Then {Z} is independent and X(t) = Z1 + - + Z. We have

lim sup X(t) = col = lim sup(X (t) X(4)) = co}nocoG 0- (Zrn+1,

Zm+27

)

for each m. So Kolmogorov's 0-1 law (Theorem 1.14) says that this event has probability 0 or 1. Since the probability is not less than 1/2, it must be 1. By the symmetry implied by Theorem 5.4(i), (5.6) is automatic from

(5.5).THEOREM 5.6 (Behavior for small t). (d =1) Let

To (w) = inf{t >0: X(w) > 0}, T4(w) = inf{t > 0: X(w) 0 for infinitely many n = P[Y(t) > 0 for infinitely many ri]= P[X(t-1 ) > 0 for infinitely many n ] = 1.This shows (5.7). The symmetry leads (5.7) to (5.8). THEOREM 5.7 (Non-monotonicity). (d = 1) Almost surely there is no interval in which X(t,w) is monotone. Proof Let [a, C [0, co). Using the set

no

in (2) of Definition 5.1, let

A[a'61 -= {w E

X(t,w) is increasing in t E [a, bil.

We claim that A[a,b1 is an event with probability 0. For t,k = a+ k(ba)ln, let

A nA =

ILO

G

9 0 X(tn,k-1,W) X(tn,k, W)},

26

1. BASIC EXAMPLESco

ThenAtalbj

=

n=1 k=1

nn An,k E

n

'7"

Since

P[An,k ] = P[X(tn,k) X(t.,k-i) 0] = and {An,k : k = 1, , n} is independent, we have P tnkn=i An,k1 = 2. Hence P[ii[a,bi] = 0. The set of w E 9,0 such that X(t,w) is increasing in some interval is the union of AM with a, bEQn [0, co), a < b, and hence, has probability 0. Similarly the set of w E 9,0 such that X(t, w) is decreasing in some interval is of probability 0.THEOREM 5.8. Let {X(t)} be a d-dimensional Brownian motion. Fix t > 0 and let

(5.9) (5.10)Let

An

:

O 4,4)

tn7 i

- " t n,N(n) t

be a sequence of partitions of [0, t] such that

mesh(n) =

1 a] > 0 for every a> 0, then there is a sequence {an} increasing to co such that the probability that Zn/an tends to 0 is 0.

30

1. BASIC EXAMPLES

E 6.18. Let {Xn} be a sequence of independent random variables on R such that Xn + 0 in prob. Let {an } be a sequence increasing to co. Then, can we say that X n 1 an > 0 a. s.? Notes Basic original references on Lvy processes and additive processes are Lvy's two books, [288] and [289]. Skorohod's [433] and [435] are the first and the second edition of a book, but each of them is rich in contents with its own merit. Chapter 4 of Gihman and Skorohod [148] is similar to a part of them. Although published only in Japanese, It's books [201], [206] should be mentioned as rigorous introductions to Lvy and additive processes. Bertoin's recent book [22] is an excellent monograph on Lvy processes with emphasis on path properties. Nice introductions to stochastic processes and their applications are Billingsley [27], Resnick [374], and Karlin and Taylor [236], [237]. Freedman [130] contains elementary treatment of Brownian motion; Theorem 5.7 and further related properties of sample functions are described there. Detailed exposition of Brownian motion is found in Lvy [289], It and McKean [207], Hida [181], Knight [265], Durrett [100], Karatzas and Shreve [234], and Revu z and Yor [376]. Exclusively treating Poisson processes is Kingman 1264 The name Lvy process is now used in many books and research papers. The name additive process for a process with independent increments is not widely employed at present, but it is used by Lvy [289] (processus additif) and It [203, 204]. It is in a broader sense without assuming stochastic continuity and Xo = 0; see Notes at the end of Chapter 2. Other names are differential process by Doob [92] and It and McKean [207], and decomposable process by Love [294]. Example 4.7 follows Feller [122].

CHAPTER 2

Characterization and existence of Lvy and additive processes7. Infinitely divisible distributions and Lvy processes in lawIn this chapter we define infinitely divisible distributions, determine their characteristic functions, show that they correspond to Lvy processes in law, and then prove that any Lvy process in law has a modification which is a Lvy process. So the collection of all infinitely divisible distributions is in one-to-one correspondence with the collection of all Lvy processes, when two processes identical in law are considered as the same. We also characterize additive processes in law and show that every additive process in law has a modification which is an additive process. Our method is based on transition functions of Markov processes. Denote by An * or pin the n-fold convolution of a probability measure p. with itself, that is, /I n sr, p n*

DEFINITION 7.1. A probability measure i on Rd is infinitely divisible if, for any positive integer n, there is a probability measure p. on Rd such that tc =Since the convolution is expressed by the product in characteristic functions, i is infinitely divisible if and only if, for each n, an nth root of the characteristic function il(z) can be chosen in such a way that it is the characteristic function of some probability measure.

EXAMPLES 7.2. Gaussian, Cauchy, and 6-distributions on iRd are infinitely divisible. Poisson, geometric, negative binomial, exponential, and ['distributions on R are infinitely divisible. So are the one-sided strictly stable distribution of index 1/2 on R of Example 2.13 and compound Poisson distributions on Rd . These facts are seen from the form of their characteristic functions in Section 2. That is, the nth roots of these distributions are obtained by taking the parameters appropriately. On the other hand, uniform and binomial distributions are not infinitely divisible. In fact, no probability measure (other than b) with bounded support is infinitely divisible, as will be shown in Section 24. Another proof that uniform distributions are31

32

2. CHARACTERIZATION AND EXISTENCE

not infinitely divisible is given by Lemma 7.5, because their characteristic functions have zeros. EXAMPLE 7.3. If {Xi} is a Levy process on Rd, then, for every t, the distribution of Xt is infinitely divisible. To see this, let tk = kt/n. Let = Px t and tin = Px(to_x(4_,), which is independent of k by temporal since homogeneity. Then z =

Xt = (X t, Xto ) + + (Xt,

Xth-i)

the sum of n independent identically distributed random variables. This is the beginning of the intimate relation between Lvy processes and infinitely divisible distributions. We begin with a simple lemma. LEMMA 7.4. If p i and 1.12 are infinitely divisible, then pi *IL2 is infinitely

divisible.

.Proof For each ri, ii Ai,nn and /12 = P2 7nn with some pi,n and 1.12,n . Hence pi*1j2 =By applying the lemma above to Gaussian distributions (Example 2.10) and compound Poisson distributions (Definition 4.1), we see that p, is infinitely divisible if

ii(z) = exp

(z, Az) +

z) +

fa R

1) v(dx)]

with A symmetric nonnegative-definite, E Rd , and y a finite measure. We will show, in Section 8, that the characteristic function of a general infinitely divisible distribution has a form which is a generalization of the above. The generalization consists in allowing v to be an infinite measure satisfying certain conditions. Now we will show in several lemmas that, for any infinitely divisible distribution /..t, the nth root is uniquely defined and further, for any t > 0, the t th power of p, is definable. LEMMA 7.5. If ti is infinitely divisible, then fi(z) has no zero, that is, ii(z) 0 for any z E

Proof For each n there is fz,, such that f(z) =Tin (z)". By Proposition 2.5(v) Lan (z)1 2 = I 2/ 12 is a characteristic function. Define yo(z) by

(p(z) = him I n (z)I 2 = {ns oo

1 if (z)

0,

0 if (z) = 0.

Since A(0) = 1 and 1'1(z) is continuous, cp(z) = 1 in a neighborhood of 0. It follows from Proposition 2.5(viii) that (p(z) is a characteristic function.

7. INFINITELY DIVISIBLE DISTRIBUTIONS AND LEVY PROCESSES IN LAW 33

Hence (p(z) is continuous on Rd . Hence (p(z) = 1 for all z that ii(z) z4 0 everywhere.

d which shows,

The converse of the lemma above is not true. For example, a binomial distribution with parameters n, p has characteristic function without zero if p 1/2, but it is not infinitely divisible. The next lemma is some complex analysis. LEMMA 7.6. Suppose that cp(z) is a continuous function from Rd into 0 for any z. Then, there is a unique C such that (,c)(0) = 1 and c,o(z) continuous function f (z) from Rd into C such that f(0) = 0 and ef(z) = (p(z). For any positive integer n there is a unique continuous function g(z) from d into C such that g(0) = 1 and g (z )h1 = (p(z). They are related as gn (z) = ef(z)In. We write f (z) = log (p(z) and g(z) = (p(z) 11 and call them the distinguished logarithm and the distinguished nth root of (p, respectively. Note that f(z) is not a composite function of cp(z) and a fixed branch of the logarithmic function. That is, cp(z i) = cp(z2 ) does not imply f (zi) = f (z2)More generally, we define, for t > 0, (,o(z) t = etf(z) , and call it the distinguished t th power of cp. We apply this to characteristic functions. Suppose that [i(z) 0 for all z. Then ii(z)' is defined for every t > 0, but it is not always a characteristic function as the remark after the proof of Lemma 7.5 shows. If ii(z) t is the characteristic function of a probability measure, then this probability measure is denoted by fit * or pt. Proof of Lemma 7.6. For z E Rd let Cz be the directed line segment from 0 to z:

Cz : w(t) = tz,

0 < t < 1.

Then, (p(w(t)), 0 0, there is a neighborhood V f(z 0)1 < e . of zo such that, for z G V, If (z) f(0) = 0, and ei(z) = Uniqueness. Suppose that f(z) is. cp(z). Then h(t) = Rtz) from the uniqueness of h(t), and hence :{(z) =-

The nth root. The complex nth root function of w is i w i/nei(l/n) arg w (multi-valued). Starting from this we can hold the same discussion as above to see the existence and uniqueness of gn (z). On the other hand, ef(z) /n satisfies the desired conditions. Hence g(z) = e. Lemmas 7.5 and 7.6 imply that, if pc is infinitely divisible, then, for each positive integer n, a distribution pn satisfying p = pnn is unique and Fin (z) = ii(z) 1 /n, that is, pn = p11n. However, it is known that, in general, vi *vi = v2 *v2 for two probability measures vi, v2 does not imply ii 1 = v2 (Feller [122], p. 506). LEMMA 7 .7 . Suppose that (p(z) and on (z), n -= 1, 2, ... , are continuous functions from Rd into C such that o(0) = (pi,(0) = 1 and (p(z) 0 and v)(z) uniformly on any compact set, then son(z) L 0 for any z. If (pn (z) log (z) log (p(z) uniformly on any compact set. 7.6. Proof. Look at the construction of the distinguished logarithm in Lemma D

hz (1) = f (z).

LEMMA 7.8. If { lik} is a sequence of infinitely divisible distributions and then 1.1 is infinitely divisible.

ii(z), we have iiik (z)1 2/71 Proof. We claim that i(z) 0. Since ilk (z) I 2/n for n -= 1, 2, ... as k 4 co. By Proposition 2.5(v) Pk (z)I 21' is a characteristic function. As (z) 2/n is continuous, it is a characteristic function by Proposition 2.5(viii). We have I(z)j 2 = Hence lil(z)I 2 is the characteristic function of an infinitely divisible distribution. Hence j2(z) 0 by Lemma 7.5 as claimed. Recall that the convergence ilk (z) 4 ii(z) is uniform on any compact set by Proposition 2.5(vi). It follows from Lemma 7.7 that loatk(z) log fi(z). Therefore Z) 1 In as k s co for any n. Since ii(z) lin is continuous, ii( z )h/2 is the characteristic function of a probability measure again by Proposition 2.5(viii). Hence p is infinitely divisible.-

7. INFINITELY DIVISIBLE DISTRIBUTIONS AND LEVY PROCESSES IN LAW 35 LEMMA 7.9. If p is infinitely divisible, then, for every t E [0, co), pt is definable and infinitely divisible. Proof We have a distribution pl/n for any positive integer n. It is infinitely divisible, since ii(z) lin = (fi(z) 1 / ("k) ) k for any k. Hence, for any positive integers m and n, prnin is also infinitely divisible by Lemma 7.4. For any irrational number t > 0, choose rational numbers rn approaching t. Then ii(z)r- > il(z) i and t(z)t is continuous. Hence [i(z)t is a characteristic function by Proposition 2.5(viii). The corresponding distribution is infinitely divisible by Lemma 7.8. Obviously p equals So . Now we will show the correspondence between infinitely divisible distributions and Lvy processes in law. THEOREM 7.10. (i) If {Xt : t > 0} is a Levy process in law on Rd , then, for any t > 0, Px, is infinitely divisible and, letting Px , = p, we have Pxt = [Lt . (ii) Conversely, if p is an infinitely divisible distribution on rd, then there is a Lvy process in law {X t : t > 0} such that Px 1 = P. (iii) If {X } and {Xa are Lvy processes in law on Rd such that Pxi = Px, then {X t } and {X } are identical in law.

In the theorem above, p is said to be the infinitely divisible distribution corresponding to the Lvy process in law {Xt }; conversely, {Xt } is said to be the Lvy process in law corresponding to the infinitely divisible distributionp.Proof of theorem. (i) Let {XL } be a Lvy process in law. The infinite divisibility of Px, is the same as the case of a Lvy process in Example 7.3. Let p = Px,. Since p = (Px, /j'1 , we have Px,/,, = pl/n Hence Pxmi = pm/n. If t > 0 is irrational, choose rational numbers r n such that rn t. We have Xrn > Xt in probability, hence Px(rn) Px,. Hence Px, = pt . Here we use stochastic continuity of {Xi } and Proposition 2.3. (ii) Let p be infinitely divisible. Then pt is a distribution with characteristic function etl61(z) . Hence (7.1) (7.2) (7.3)iss *yt

_ ps+t , 60

PU

as t

O.

Let us construct the corresponding Lvy process in law. Consider SZ, .T, and Xt (w) = co(t) in Kolmogorov's extension theorem 1.8. For any n > 0 and any 0 < to < ti < - < tn , define

(7.4)

(Bo x x Bn)

36

2. CHARACTERIZATION AND EXISTENCE

Pt (dYo)lsoX

-t (dY1) 1 B1(Yo + Yi) (dYn) 1 B (Yo + + Yn)

Then pto ,...,t is extended to a probability measure on B((1ler-1-1 ) and the family {u to,..} satisfies the consistency condition by (7.1). Hence, by Kolmogorov's extension theorem, we get a unique measure P on F such that (7.5)

P[Xto E B0,

, Xt..

E

B,j =

(Bo x

- - -

x

Bit).

In particular, Xt has distribution pt . Let us show that {Xt : t > 0} is a Levy process in law. If 0 < to < < in , then we have, from (7.4) and (7. 5), (7.6)

E[f (X to , . , Xt)1 =f(Yo, Yo + Yi, - - , Yo +"+ x (d n)

Atn -n-1(dyn )

for any bounded measurable function f. Let z1 ,... , z. G Rd and

f (so , . . . , xn) = exp iE(zi,xi - xi-1) ( nThen

E[exp(i. E(zi, Xt.; -

1It follows that

(do . . f exp (i (zi, yi))

(dyn )

j=1

=nf

exp(i(z.i,Y;)),ati -ti-1 (dxf).

E[exp(i(zi, Xt; - Xt; _ 1 ))] = f exp (i(zjyi))/Lti-ti-1 (dyi),which shows that Xt1 - X, has distribution ttii -ti- 1 and that

(7.7) E [exp E(zi, Xi ; - -X -t3 _1 ))] =5=1 i=i

E[exp(i(zi, Xt; - Xii_1))]-

8. REPRESENTATION OF INFINITELY DIVISIBLE DISTRIBUTIONS

37

By Proposition 2.5(iv) this says that {Xt} has independent increments. The convergence (7.3) says that Xt * 0 in prob. as t 0 (Exercise 6.8). Hence > 0 as s t. Hence we have (1), P[1X., X4 > s] = P[ (2),(3), and (4) of Definition 1.6. That is, {Xt} is a Lvy process in law.

= (iii) Let {Xt } and {Xa be Lvy processes in law and X1 --!1 X. Then, by (i), Xt Jq. It follows that X,st X3 -L X's+t X's for any t and s. 1Hence (Xt., Xt, Xt.,...,Xt

--`1 (X., Xi

XL_ 1)

for 0 < to < < tn. by independence. Since (X, .. , Xt,,) is a function of (Xt., Xt, Xt., , Xt X1,,), we get

(X10 , . .completing the proof.

Xt)

P4o , - )q),

REMARK 7.11. Even if {XL } has stationary independent increments and starts at the origin, the assertion (0 of Theorem 7.10 is not true unless {Xt } is stochastically continuous. In this case, the distribution of Xt is infinitely divisible but is not always equal to p i . For example, let f (t) be a function such that 1(t) + f (s) = f (t + s) for all nonnegative t and s but that 1(t) is not a constant multiple of t, and let Xt = f (t). Such a function is given by G. Hamel [162].

8. Representation of infinitely divisible distributionsThe following theorem gives a representation of characteristic functions of all infinitely divisible distributions. It is called the Lvy-Khintchine representation or the Lvy-Khintchine formula. It was obtained on R around 1930 by de Finetti and Kolmogorov in special cases, and then by Lvy in the general case. It was immediately extended to i . An essentially simpler proof on R was given by Khintchine. This theorem is fundamental to the whole theory. Let D = Ix: 'xi < 11, the closed unit ball. THEOREM 8.1. (i) If II is an infinitely divisible distribution on r,

then

(8.1)

ii(z) +fRd

1 i(z, x)1 D (x))1/(dx)] ,

z E

Rd,

where A is a symmetric nonnegative-definite d x d matrix, y is a measure on Rd satisfying

(8.2)

v({0}) = 0

and

L as12

A 1)v(dx) 0, as 'xi > co.

Then (8.1) is rewritten as

(8.5)

[i(z) = exp [

Az) + i(7,, z)

+ f(ei (z* 1 i(z, x)c(x))v(dx)] Rdwith 7e G Rd defined by 1 D (x))v (dx) Rd (Here it is enough to assume c(x) = 1 + O(Ixj), l xi > 0, instead of (8.3), but we will use (8.3) in Theorem 8.7.) The following are examples of c(x) sometimes used:

(8.6)

-Ye =

f s(c(x)

c(x) = 1I 0,

c(x) =11 (1 + 1x1 2), c(x) = 11*.11. (ei(x* 1 i(z,x)1 D (x))v(dx)].

Since the measure v restricted to {Ix! > 1/n} is finite, this on (z) is the convolution of a Gaussian and a compound Poisson distribution and, hence, is the characteristic function of an infinitely divisible distribution (Example 7.2 and Lemma 7.4). As n co, it converges to (p(z). On the other hand, (p(z) is continuous, as is noticed at the beginning of the proof of part (ii). Hence o(z) is the characteristic function of an infinitely divisible distribution by Proposition 2.5(viii) and Lemma 7.8. The result below incorporates an essential part of the proof of part (i). Let us write f E if f is a bounded continuous function from Rd to R vanishing on a neighborhood of 0.

co

THEOREM 8.7. Let c(x) be a bounded continuous function from R d to R satisfying (8.3) and (8..4). Suppose that p (n = 1, 2, ... ) are infinitely divisible distributions on Rd and that each A.7,(z) has the Lvy-Khintchine representation by generating triplet (A,1,,,, On) c . Let p be a probability measure on Rd . Then p. p if and only if p is infinitely divisible and pi(z) has the Levy-Khintchine representation by the generating triplet (A, u, 0) c with A, v, 13 satisfying the following three conditions.

(1) If f E

Co,

then

n--.c Rd

lim f f (x)v n (dx) = f f (x)v(dx). Rd

42

2. CHARACTERIZATION AND EXISTENCE

(2) Define symmetric nonnegative-definite matrices Aft,, by

(z , An, e z) = (z , A n z) + f (z , x) 211n(dx)isiEThen

lim lim sup 1(z,el()n

- (z, Az)I = 0

for z E Rd .

(3) On If we use (8.1) for the Lvy-Khintchine representation, Theorem 8.7 cannot be proved. This is because of the discontinuity of 1D (x). Proof of theorem. Assume that pn, p. Then, p is infinitely divisible (Lemma 7.8) and /7i(z) 0 (Lemma 7.5). It follows from Lemma 7.7 and Proposition 2.5(vi) that

(8.11)

log

(z) -+ log (z)

tmifounly on any compact set.

Define pn (dx) = (1x1 2 A 1)vn (dx). We claim that {pn } is tight in the sense that

(8.12) (8.13)

sup p(r) 0 as the conclusion is evident in the case pn (litd) + O. Define v by v({0}) = 0 and v(dx) = (1x1 2 A 1) -1 p(dx) on {Ix' > 0 } . The measure p may have a point mass at 0, but it is ignored in defining v. Let-

(8.14) (8.15)

g(z , x) = el(z'x)

-

1

-

i(z, x)c(x),

which is bounded and continuous in x for fixed z. We have log rin (z) = = wheren ,E

(z, A,z) + i(/3,., z) + f g(z,x)v n (dx)(z, A n,,z) + z) + fl E + 4E,,

=(g(z ,x) + (z, x) 2 ) (ix 12 A 1.) -1 pn(dx),IXI5E

=

f ix1>E

g(z, x)(1x1 2 A 1) -1 pn (dx).

8.

REPRESENTATION OF INFINITELY DIVISIBLE DISTRIBUTIONS

43

Let E be the set of e > 0 for which fix1=5 p(dx) = 0. Then-1 2 f Jnk & = ixi>, g(z, x) (IX' A 1) p(dx),

for e G E.

Hence

(8.16)Furthermore we get

E;10 k000

rim fim

=Jga

g(z,x)v(dx).

(8.17)

lim sup j/n, e l = 0 do ,'

from (8.12) since (g(z, x) + (z, x) 2 )(1x1 2 A 1)' tends to 0 by (8.3) as x > 0. Considering the real part and the imaginary part of (8.15) separately and using (8.11), (8.16), and (8.17), we get

(8.18) (8.19)

lim lim sup(z, = urn lim inf (z, An,dz), EDE10 k-- s co EDdio lim sup(0,, z) = lim inf (0,, z),k.co

and both sides in (8.18) and (8.19) are finite. By (8.19) there is 0 such that + 0. Since each side of (8.18) is a nonnegative quadratic form of z, it is equal to (z, Az) with some symmetric, nonnegative-definite A. In (8.18) we can drop the restriction of e to the set E, because (z, is monotone in e. It follows that ii(z) has representation (8.5) with these A, v, and 0 (in place of 7,) and that (1), (2), and (3) hold with n co via the subsequence {it,}. The A, 1/, and 0 in the triplet (A, u,/3) are unique, because we already proved part (ii) of Theorem 8.1. As we can begin our discussion with any subsequence of {tin }, this uniqueness ensures that (1) and (3) hold for the whole sequence Now, looking back over our argument, we see that

(8.20)

lim lim sup (z , 24,,Ez) = lim lim inf (z, A.,,z) = (z, Az).EI 071,co

E10

fl-.00

This is equivalent to (2). This finishes proof of the 'only if' part, provided that (8.12) and (8.13) are true. Proof of (8.12) and (8.13) is as follows. Let [h, i]" = C(h). We have

(8.21)

f log Tin (z)dz'(h)

=

f C(h)

(z, A n z)dz ltd

14,(dx) f g(z,x)dzC(h)

d f (i [

vn(dx). (2h) _ fl sin h.xj) hxi

d

44

2. CHARACTERIZATION AND EXISTENCE

The leftmost member of (8.21) tends to fc(h) log [i(z)dz as n ting h = 1, and noting

co. Let-

inf (1

ri sin xi )

d

(le A 1) -1- > 0,

we see that (8.12) is true. Since

limh10

1

(2h)d

fC(h)

log t(z)dz = 0,

it is shown that, for any

E>

0, there are no and ho such thatitoXj

Ld ( 1d

d sin hoxi)

1I

vn (dx) no.

If 'xi > 2-4/h0 , then ix30 1 > 2/h0 for some jo and

1Hence

nj=1

sin floXi >

1

sin hoxi,

h.ox3

hoxiovn (dx) 1

1 1 > jx3-o I 2

E

for n > no .

1.1>2../alho

Hence

pn (dx) < efix! >21,/a/ho

for n > no.

This proves (8.13). Let us prove the 'if' part. Define pn (dx) = ( 1x12 A 1)vn (dx) as above, and p(dx) = ( 1x12 A 1)v(dx). Let the set E be as above. Then we get (8.16) from condition (1). Since conditions (1) and (2) imply uniform boundedness (8.12) of Ipnl, we get (8.17) also. Hence, using (2) and (3), we have

lim log rin(z) =IL 0 CO

Az) + i(0, + g(z,x)v(dx). D

The right-hand side is equal to log ii(z). Therefore /i n > p.

Now, using the 'only if' part of Theorem 8.7, completion of the proof of Theorem 8.1 is easy. Proof of Theorem 8.1(i). We are given an infinitely divisible probability measure p. Choose tn 0 arbitrarily. Define /i n by

An(z) = exp[t; 1 (1.7(z) t- 1)] exp [t,V.exp [t,T1(etn l05

f

Rdvol

(ei(;s)

1),atn(dx)].

The distribution p, is compound Poisson. Note that

(z)

i(z)

1)] =

exp[t; 1 (tn log ii(z) -I- 0 (t2.))]

8. REPRESENTATION OF INFINITELY DIVISIBLE DISTRIBUTIONS

45

ekigi4z) = "ri(z). Since fin has the for each z as n - oo. Hence fiyi (z) representation (8.5) in Theorem 8.7, we can apply the theorem and conclude that Ti(z) has the Lvy-Khintchine representation with triplet (A, ii, p),. This representation can be written in the form (8.1).COROLLARY 8.8. Every infinitely divisible distribution is the limit of a sequence of compound Poisson distributions.

Proof See the proof of Theorem 8.1(i).COROLLARY 8.9. Let tn J. O. If v is the Lvy measure of an infinitely divisible distribution i then, for any f E,

rig

f

Rd

f (x),u tn (dx)

f f (x)v(dx). Rd

Proof The distribution fin in the proof of Theorem 8.1(i) has Lvy measure [tyi liiinhzdvol. Condition (1) in Theorem 8.7 gives the result. ElEXAMPLE 8.10 (r-distribution). Let ti be a r-distribution with parameters c, a as in Example 2.15. Let us show that

(z) = exp [c f

- 1)

e -CYX

.

This is representation (8.7) with A = 0, v(dx) = c1(003)(x)x-1 e-a'dx, and -= 0. This fi is not a compound Poisson distribution, because v has total mass co. By (2.15) and by

log(1 + & 1 u) =

f f

dy

= f u dy f oe edx + y o o

e-"dx f e-Yxdy, e -oa - 1) - dx].by ana-

the Laplace transform is expressed as

L(n) = exp [cf

Extending this equality to the left half plane fw E C: Re w < lyticity inside and continuity to the boundary, we get

ens(dx)Let w = iz, z E IR to get i(z).,

ax eexp[c f ( wx - 1) dx] . e x

EXAMPLE 8.11 (One-sided strictly i-stable distribution). The Laplace transform of the fi of Example 2.13 is

L(n) = e- c`/7` = exp c(27) -112o

(e-u'T 1)X-3/2 dX

U

>0.

46

2. CHARACTERIZATION AND EXISTENCE

The last equality is obtained as follows: ... (e 1)x -312dx = f x-3/2 dx o . --= u f e'Ydy

fo

. ue'Ydy o OE, x -372 dx

JyCO

= 2u f

e'Yy -1/2dy = 2u 112 r() =

Extending the expression to the left half plane, we get, on the imaginary axis,CO

fi(z) = exp [c(21-) -1/2 f

(eizz 1)x -3/2dx ,

z

E R.

This is the form (8.7) with A = 0, v(dx) = 1(o,00)(x)c(270 -1/2x-3/2ax and ', "Yo = 0. Again p is not a compound Poisson distribution. REMARK 8.12. All infinitely divisible distributions in Example 7.2 are such that their infinite divisibility is obvious if we look at explicit forms of their characteristic functions. That is, their nth roots in the convolution sense are obtained by taking their parameters appropriately. But there are many other infinitely divisible distributions whose infinite divisibility is more difficult to prove. We list some such distributions on R with the papers where their infinite divisibility is proved. Here c's are normalizing constants. We have chosen scaling and translation appropriately. Student's t-distribution (Grosswald [155], Ismail [196]) p(dx) = c(1 + x2 ) -(a+1)/2 dx, Pareto distribution (Steutel [441], Thorin [473 ]) p(dx) = a E (0, oo);

c1(0,03) (x)(1

+

a E (0, co);

F-distribution (Ismail and Kelker [197]) p(dx) = c1(o, c0)(x)x'3-1 (1 + x) -Q-Pdx, a, p E (0, co);

Gumbel distribution (extreme value distribution of type 1 in [218]) (Steutel [442]) P( 00 , = Weibull distribution (extreme value distribution of type 3 in [218]) with parameter 0 0;

9. ADDITIVE PROCESSES IN LAW

47

log-normal distribution (the distribution of X when log X is Gaussian distxibuted) (Thorin [474]) P(dx) = 1 (30,) (x)x -l e-' 00gx)2 dx, a E (0, 00 );

logistic distribution (Steutel [443]) x] = (1 + e') -1 for x E R; half-Cauchy distribution (Bondesson [47]) p(dx) = 27-1 1 (0,co) (x) (1 + x2 ) -1 dx. See Exercise 55.1 for Pareto and Weibull with 0 < a < 1. Other such examples are mixtures of Ildistributions of parameter c when c is fixed in (0,2] (Remark 51.13). There are many infinitely divisible distributions on R with densities expressible by Bessel functions (Feller [122], Hartman [163], Pitman and Yor [339], Ismail and Kelker [197], Yor [525], see also Example 30.11, Exercises 34.1, 34.2, and 34.15). On Rd the following are known to be infinitely divisible (Takano [454, 455]): (8.22) (8.23) tt(dx) = ce- lxidx; 1512)-.-(d/2) , P(dx) = c(1

a E (0, oo).

9. Additive processes in law Infinitely divisible distributions have close connections not only with Lvy processes but also with additive processes. THEOREM 9.1. If {X,: t > 0} is an additive process in law onRd, then, for every t, the distribution of X, is infinitely divisible. We prove this theorem from the next result. This is one of the fundamental limit theorems on sums of independent random variables, conjectured by Kolmogorov and proved by Khintchine. DEFINITION 9.2. A double sequence of random variables {Z,,k: k = 1,2, ,r,; n = 1, 2, ... } on Rd is called a null array if, for each fixed n, Zn2, - 7 Znr are independent and if, for any r>0, (9.1) The sums Sn = lim max P[ IZnk >1 0} be a Lyy process on Rd. Then, for any s > 0, X s : t > 0} is a Lyy process identical in law with {X t : t > 0}; -PCs+t Xe : t > 01 and {X t : 0 < t < s} are independent. Proof. Fix s and let 4 = X,s+t X. Since Z0 = 0 and Z 2 Zt, = X3+t2 X 3+11, the definition of the Lvy process for {X 1} implies that {Zt } is a Lvy process. The rest of the assertion is immediate. D REMARK 10.8. Sometimes it is useful to consider a random starting point. Given a transition function Ps,t(x , B), let {Yt } and (0, F , Pci,a) be as in Definition 10.2. Using Proposition 1.15, we can prove from Proposition 10.6 that po,a [A] is measurable in a for any A E j0 For a probability measure p on Rd, define Pc4P[A] = p (d a)PQa[A]. A stochastic process {Xt : t > 0} defined on a probability space (S-2, 1 , P) is called a Markov process with transition function Ps,t(x, B) and initial distribution p, if it is identical in law with the process {Yi } on the probability space (SP, J=6,P).

11. EXISTENCE OF LEVY AND ADDITIVE PROCESSES 11. Existence of Lvy and additive processes

59

We have shown that, for any infinitely divisible probability measure it, there exists, uniquely up to identity in law, a Lvy process in law with distribution /2 at time 1. Now we will show that any Lvy process in law has a Lvy process modification, which establishes the correspondence between the infinitely divisible probability measures and the Lvy processes. More generally, we will deal with additive processes in law. For this purpose we will give a sufficient condition for a Markov process to have sample paths right-continuous with left limits This is a result of Dynkin [106] and Kinney [263]. A sufficient condition for sample path continuity is also given, which proves the existence of the Brownian motion. Denote the c-neighborhood of x by DE (x) = fy:ly xi < Eh and its complement by De (s)e. Suppose that we are given a transition function Ps,t(x,B) on . Let (11.1) as,T(u) -= sup{/ 3 ,(x, 1),(x)c): x E Rd and

s, t E [0, T1 with 0 < t s 0} be a Markov process on Rd defined on (52, .1, P) with transition function (z , and a fixed starting point. If (11.2) lim ct,,T(u) = 0

u.Lo

for any E. > 0 and T > 0,

then there is a Markov process {X t > 0} defined on the probability space q: (SZ,F,P) such that (11.3)

P[Xt = Xn =1 for t > 0,

and )q(co) is right-continuous with left limits os a function of t for every w. This VD. automatically satisfies (11.4)

PfX;= 1

=- 1

for t > O.

If, moreover, the transition function satisfies (11.5) urn a, T (u) = 0 uio u for any c

>0

and T> 0,

then there is Il E F with P[121 ] = 1 such that, for every w G 91, -,q(w) is continuous as a function of t. Let M c [0, oo) and c > 0. We say that .)et (w), with w fixed, has eoscillation n times in M, if there are to, t1, , tn in M such that to 0} is identical in law with the process {a 112 Xt : t > 0} (Theorem 5.4). This means that any change of time scale for the Brownian motion has the same effect as some change of spatial scale. This property is called selfsimilarity of a stochastic process. There are many selfsimilar Lvy processes other than the Brownian motion; they constitute an important class called strictly stable processes. Stable processes are a slight generalization; they are Lvy processes for which change of time scale has the same effect as change of spatial scale and addition of a linear motion. Stable distributions were introduced in the 1920s by Lvy and stable processes have been extensively studied since the 1930s. In this chapter we give representations of stable and strictly stable processes. Further we determine selfsimilar additive processes. Extension of the notion of selfsimilarity to semi-selfsimilarity is also studied. Semi-stable, selfdecomposable, and semiselfdecomposable distributions appear. In this section, we will define stable, strictly stable, semi-stable, and strictly semi-stable distributions first, and then define processes with these names as Lvy processes corresponding to these distributions. On the other hand we will define selfsimilarity, broad-sense selfsimilarity, semiselfsimilarity, and broad-sense semi-selfsimilarity for general stochastic processes, and prove the existence of the exponent a quantity that expresses the relationship between the change of time scale and that of spatial scale. These notions applied to Levy processes give equivalent definitions of the stable processes and the like. DEFINITION 13.1. Let p be an infinitely divisible probability measure on d . It is called stable if, for any a> 0, there are b> 0 and c E Rd such that (13.1)[i(z)a t

ri( bz) ei(c,z) .

It is called strictly stable if, for any a> 0, there is b> 0 such that (13.2)

ii(z)a =69

70

3. STABLE PROCESSES AND THEIR EXTENSIONS

1, there are b > 0 and It is called semi - stable if, for some a> 0 with a c E Rd satisfying (13.1). It is called strictly semi - stable if, for some a > 0 with a 1, there is b> 0 satisfying (13.2).DEFINITION 13.2. Let {Xt : t > 0 } be a Lvy process on Rd. It is called a stable, strictly stable, semi-stable, or strictly semi-stable process if the distribution of Xt at t = 1 is, respectively, stable, strictly stable, semi-stable, or strictly serai-stable. EXAMPLE 13.3. If p is Gaussian on I, then ii(z) = and p is stable, as it satisfies (13.1) with b = a112 and c = (_611/2 Thus, if p is Gaussian with mean 0, then it is strictly stable. If p is Cauchy on with parameter c > 0 and -y e Rd (Example 2.12), then ii(z) = e'lzi+i("YM and p is strictly stable, satisfying (13.2) with b = a. If

(13.3)

II

= E 714:' 6

co

bn X0

with b> 1, 0 27r1z1 -1 . This is absurd, as Izi can be chosen arbitrarily small D LEMMA 13.10. Let Z and W be non-constant random variables on Rd. Let Zn be random variables on Rd, b > 0, and c E Rd. If Pz. P and z Pb,,Z-Fen Pw as n > co, then ba b and c c with some b E (0, co) and c E Rd as n co, and bZ + c W. Proof Write Pz pn, Pz = p, and Pw = p. Then tg-(z) > 1 -i(z) and ii,j (bnz)e*--z ) Rz) uniformly on any compact set. Let boo be a limit point of { bn} in [0, co]. If be,, = 0, then, letting n co via the subsequence nk satisfying bn, b, we get Iii(0)1= 1, which shows that 1j- 1z)1 =- 1 and Z is constant by Lemma 13.9, contradicting 9 the assumption. If boo = co, then Vink (z)1 = (bnb7 k1 z)1 ) 1(0)1 -= 1 7 and hence 1/.7(z)1 = 1, contradicting the assumption again. It follows that 0< < co. There is e> 0 such that ii(booz) 0 for lzl < E. It follows that

R z)/ ii (boa z)

13. SELFSIMELAR AND SEMI-SELFSIMILAR PROCESSES

73

uniformly in z with zj be a broad-sense semi-selfsimilar, stochastically continuous, non-trivial process on IRd with X0 = const a.s. Denote by r the set of all a > 0 such that there are b> 0 and c(t) satisfying (13.5). Then: (i) There is H > 0 such that, for every aEF,b= aH (ii) The set r n (1, co) is non-empty. Let (to be the infimum of this set. If a0 > 1, then F = {aon: n E Z } and {X 2 } is not broad-sense selfsimilar. If ao = 1, then F = (0, co) and {X 2 } is broad-sense selfsimilar. Proof. By Lemma 13.8, b and c(t) in (13.5) are uniquely determined by a. We write b = b(a) and c(t) = c(t, a). The set F has the following properties.E r and b(1) = 1. (2) If a E F, then a -1 E F and b(a 1 ) = b(a) 1 . (3) F n (1, co) is non-empty. (4) If a and a' are in F, then aa' E I' and b(aa') = b(a)b(d). (5) If an E F (n 1, 2, ... ) and an a with 0 < a < co, then a E F and b(an ) b(a).

(1) 1

The property (1) is obvious. If (13.5) holds, then

{X(a -l t)}

{b -l X(t)

Thus (2) holds. Since F contains an element other than 1, the property (3) follows from (2). If a and a' are in r, then

(13.7)

{X(aa' t)}

{b(a)X(a' t) + c(a't,a)} {b(a)b(d)X(t) + c(a't, a) + b(a)c(t, a')},

which shows (4). To prove (5), write bn = b(an ) and cn (t) = c(t, an). Then d X 1,2 bnX i + cn (t) and Xt Xot in prob. If t is such that Xot is non-constant, then Xt is non-constant and, by Lemma 13.10, bn b and cn (t) c(t) for some b E (0, co) and c(t) E Rd . Since such a t exists by the non-triviality of {X2 } , we have bn 4 b. Now, for any t, the last part of the proof of Lemma 13.10 shows that cn (t) tends to some c(t). Hence d Vat/ = {bXt + c(t)}, which shows (5).

74

3.

STABLE PROCESSES AND THEIR EXTENSIONS

We denote by log F the set of log a with a G F. Then, by (1)-(5), log r O. Denote the is a closed additive subgroup of R and (log n n ((), co) infunum of (log 11 n (0, co) by ro- Suppose that ro > O. Then we have ro E log r and roZ -= { ron: n E Z} C log F. If there is r E (log r) (roZ), then nro 0, then log r = roZ. If ro = 0 and there is r in R \ (log F), then we have (r + e) C R\ (log r) with some e > 0 by the closedness of log r, and, choosing s E log r satisfying with some n G Z and 71.5 G log r, which O < s < 2e, we get r-e 1. In fact, suppose that a> 1, a E r, and b(a) 1 and a G = fit (b(a)nz)ei(z'c(t'an' Hence ce=t(b(a)'z) I = Iiit (z)I

Then

for n E Z, z E Rd.

for n E Z, z E Rd .

1 uniformly in w in any compact Since X(0) is constant, we have (w) -cc. Since lb(a)z < IzI for n < 0, we have set as n 0 5_ 1 - liZe=t(b(a) -n z)I 5_ supivIzI

- Iricon (w) I)

0

as n -4 -co. It follows that (z) I = 1 and hence Xt is constant by Lemma 13.9. Since t is arbitrary, this contradicts the non-triviality. This proves (6). Now we prove the assertion (i). Suppose that ao > 1. Let H = (log b(a0))/ (log ao). Then H > 0 by (6). Any a in F is written as a = aon with n E Z. Hence b(a) = b(ao) = aoHn = a 1 . In the case ao = 1, we have F-= (0, cc) and the properties (4) and (5) yield the existence of H E R satisfying b(a) aH . Also in this case, the property (6) shows that D H > O. This proves (i). DEFINITION 13.12. The H in Theorem 13.11 is called the exponent of the non-trivial broad-sense semi-selfsimilar process. It is uniquely determined by the process. If a is in P n (1, co), then a and aH are called, respectively, an epoch and a span of the process. Instead of broad-sense semi-selfsimilar with exponent H, we sometimes say broad- sense H-sentiselfsimilar. The semi-selfsimilarity implies the broad-sense semi-selfsimilarity. Thus we say that MI is semi-selfsimilar with exponent H or H - semiselfsimilar if it has exponent H as a broad-sense semi-selfsimilar process and if it is semi-selfsimilar Similarly we use the words broad- sense H-selfsintilar and H-selfsimilar.

13. SELFSIMILAR AND SEMI-SELFSIMILAR PROCESSES

75

REMARK 13.13. Let {Xi} be a semi-selfsirnilar, stochastically continuous, non-zero process on Rd with X0 = 0 a. s. Let r = la > 0: there is b> 0 satisfying (13.4)1. Then we can prove the statements (i) and (ii) of Theorem 13.11 with broad-sense semi-selfsimilar replaced by semi-selfsimilar. The H thus determined equals the exponent in Definition 13.12, if {Xi } is non-trivial. If {Xi} is a non-zero, trivial, semi-selfsimilar, stochastically continuous process with X0 = 0 a. s. and if a is an epoch, then Xi = tHg(logt) a. s., where g is a continuous periodic function with period log a. Also in this case, H is called the exponent.

Broad-sense selfsimilar processes are related to selfsimilar processes in the following way. PROPOSITION 13.14. If {X t } is broad-sense selfsimilar, stochastically continuous, non-trivial, and X 0 = const a.s., then there is a continuous function k(t) from [0, co) to R d such that {X t k(t)} is selfsimilar and X 0 k(0) = 0 a. s. Proof. By Theorem 13.11 {Xi } has the exponent H > O. For any a> there is unique ca(t) such that {Xat } = d Xt+ ca (t)}. If a > ao > 0, then ea(t) is convergent, because Xai and aH Xt are convergent in probability. It follows that ca (t) cao (t) as a * ao > O. As a 1, 0, c,. (t) tends to X0 . So let co (t) = X0 . We have c(t) = ca(dt) + aH ca, (t) for a> 0, > 0, as in (13.7). Now let k(t) = ci (1). Then k(t) is continuous on [0, co) and {Xai k(at)}

X t + ca(t) c at (1)} = X t a"c(1)} = {a"(X t k(t))},

which is the desired property.Proposition 13.14 can be extended. The statement remains valid if we replace

selfsimilar by semi-selfsimilar (Maejima and Sato [3001).THEOREM 13.15. Let {X t : t > be a non-trivial semi-stable process on Rd with exponent H as a broad-sense semi-selfsintilar process. Then H > 1/2.

Proof By Proposition 13.5 {Xi } is broad-sense semi-selfsimilar and, by Theorem 13.11, the exponent H > 0 exists. We have (13.6) with b = aH , where a> 1 is an epoch and b> 1 is a span. Let the generating triplet of {Xi } be (A, v, -y). Then that of {X at} is (aA, av, a-y). We define, for any r > 0, a transformation T,. of measures p on Rd by (13.8)

(Tf p)(B) = p(r-1 B)

for B E B Rd).(

76

3. STABLE PROCESSES AND THEIR EXTENSIONS

Then, by using Proposition 11.10, we see that the generating triplet of {a/I Xt+tc} is (a2H A, TiY,1(a)) with some -y(a) E Rd and b = . Therefore, by the uniqueness, aA = a2HA and av = Tbv. By the non-triviality, we have O. It follows 0, then H = 1/2. Suppose that v A 0 or v O. If A from av = Tbv that a- lv = Tb-lu. Iteration gives

(13.9)Let

any = Ttpv

for n E Z.

(13.10)

Sn (b) =-- {X E Rd :

< lx1 < bn-1-1 }

for n E Z.

Then Sn (b) = So (b) and v(S,z(b)) = (Tb--v)(So(0) = ci-n v(So(b)). The set {x: 0 < I xi < 1} is partitioned into S_n_i(b), n E Z+, and the set O. {x: lxi > 1} is partitioned into Sit (b), n E Z+ . It follows that v(So(b)) Since (13.9) is equivalent to an f 1B(x)v(dx) = f 1 B (bnx)v(cix), we have

f f (x)v(dx) = f f (ba x)v (dx)for any nonnegative measurable function f. Thus(dX) (b)IX12 / (dX) = f 1 So (b)(b n X)IXI 2 /

=a

1so(b)(x)ibnxi21/(dx)

= a

- 2H ) 15..(b) i xi 2 1/( dx).co, that is, H >

Since fiz1 1 of the process is also called a span of ji. Sometimes we say a-serai-stable instead of semi-stable with index a. The definition of the index applies to stable processes and distributions, as they are special cases of semi-stable. We say a-stable for stable with index a. By Theorem 13.15 the index a satisfies 0 1/2, that is, 1, S(b) be the set in (13.10). The transformation T,. of measures is defined for r> 0 as in (13.8). Note that, if p is a measure concentrated on Sn (b), then Tb.p is concentrated on Sii+m (b). If p is the distribution of a random variable X, then Trp is the distribution of rX. The restriction of a measure p to a Borel set E is denoted by [METHEOREM 14.3. Let p be infinitely divisible and non-trivial on Rd with generating triplet (A, u, 'y). Let 0 < a < 2. (i) Let b> 1. Then the following three statements are equivalent:

so.

(1) p is a-semi-stable with b as a span; (2) A =- 0 and(14.1) v = b'Tbv;(3) A = 0 and, for each integer n, the measure u on Sn (b)is determined by the measure v on So (b) by

(14.2)

kils(b)

78

3. STABLE PROCESSES AND THEIR EXTENSIONS (ii) The following statements are equivalent:

(1) p is a-stable; (2) A = 0 and(14.3)u = b"Tbv for every b > 0; (3) A = 0 and there is a finite measure on S such that

(14.4)

v(B) = f (d)

1B (r)

_ rdro, i+

for B e B(R d).

Proof. (i) Suppose that (1) holds. Consider the corresponding semistable process. Since 0 < a < 2, the proof of Theorem 13.15 shows that A = O. Thus v 0, since p is non-trivial. The Lvy measure I/ satisfies av = Tbv with a = , as in the proof of Theorem 13.15. That is, (2) holds. The condition (2) implies (3), since we have (13.9), that is,

b"v(B) = (Tbnv)(B) = v(b'B)

for B e

me), n G Z,

which gives (14.2) for B C S n (b). Now suppose that (3) holds. For any B E (Rd) , let Bn = B n Sn (b). Then, by (14.2),

v(B) =

Ems0,(Bn) = Erna[v]so(b) (b-nB.) nez nez E b-"v(b'B n So(b)).nEZ

Therefore

(Tbv)(B) = v(b -1 B)

= Eb-"v(b-n-iB n So (b))nEz n So (b))-

r E b"v(b'BnEZ

Hence we get (2). Let us see that (2) implies (1). Consider a random variable X whose distribution is p. Then A(bz) is the characteristic function of the distribution of bX . By Proposition 11.10, the distribution of bX has generating triplet (0, Tbv,7(b)) with some -y(b) G Rd . On the other hand, fi(z)a is the characteristic function of the distribution with generating triplet (0, ay, cry). Hence (z)a = fi(bz)ez) with a -= if and some c, that is, p is a-semi-stable having b as a span. (ii) Suppose that (1) holds. Then p is a-semi-stable and any b> 1 can be its span. Hence, by (i), (14.1) holds for any b> 1. Since the property v(B) = b'v(b'B) for all B E B(Rd) implies v(bB) = b"v(B) for all B e B(R d), (14.1) remains true with b replaced by b-1 . Hence we have (2). Assume the condition (2). Let us write, for E c (0, cc) and C C S,-

(14.5)

EC ---- {x E li \ 101: x E E and xx E C}.

14. REPRESENTATION OF STABLE AND SEMI-STABLE

79

Define a finite measure A on S by

(14.6)

A(C) = av((1, co)C)

for C E B(S).

Define //(B) by the right-hand side of (14.4). Then is a measure on Rd v'({0}) -= 0 and, for b> 0 and C E B(S), with

v' ((b, eo)C) = A(C) f

dr X(C) = b'v((1, co)C) = b r " = v(b(1, co)C) = v ((b, oo)C)

by (14.3). It follows that v' (B) = v(B) for all B G /3(1Rd \ 101). (Here we have used Proposition 1.15. Fix e> 0 and consider the set fx: 1x1 > el. Let be the collection of all sets of the form (b, )C with b > e and C E B(S). Use of Proposition 1.15 leads to the conclusion that v' = 1/ on cr(.,45) , where o(A) is the collection of all Borel sets in fx: 1x1 > el. It follows that i/ = v on /3(Rd \ {O}).) Thus we have (3). If we assume (3), then we get (2), since we obtain (14.3) from (14.4). Assume the condition (2). By (i), p is a-semi-stable and any b > 1 can be chosen as a span. That is, for any b > 1, there is c E Rd such that ii(bz)ei (c,z) . Since z is variable, it follows that

=for any b> 1. Hence p is a-stable.

z ) el(-b-*-1c,z)

0

REMARK 14.4. In Theorem 14.3(ii), the measure A on S is uniquely determined by p, because (14.4) implies (14.6). We call any positive constant multiple of A a spherical part of the Lvy measure v. For any non-zero finite measure A on S and for any 0 2 0 (1,0) as t -> oo. The asymptotic expansions of pa (1, x) are obtained by Linnik 12911, Skorohod [431], and others. We give, without proofs, the results (with misprints corrected and with some formal changes) in Zolotarev [536]. We can fix the parameter c without loss of generality. Assume that c equals cos(), 1, or cos(n-0 2 ) for a G 1, = 1, or > 1, respectively. Let a' = 1/a. Let p = (1+ 0)/2 or = (1 - /3 -')/2, according as a < 1 or > 1. The following (i)-(iii) are representations of p(1, x) by convergent power series. (1) If a > 1, then-

(14.30)

pO (1, x) = 1

c0 (-1) .-1 F( na' -1- 1) . ks rup)x n - 1 inIt!

for x E R.

r n=1

(ii) If a < 1, then

(14.31)

p (1, x) = 1

c0 (-1) , 1 F(na -I- I) (sin rnpa)x ,,,,i n!

for z> 0.

(iii) If a = 1 and /3> 0, then co po( i, x) _ 1 E (_ir-inbnxn-i (14.32) where

for x e R,

b. =

1

fr exp (-Mi log u)un-1 sin (i(1 + Mu)du.

The following (iv)-(vii) hold for any positive integer N. (iv) When a < 1, 0 1, x E R, and x 0, (14.33) po ( 1, x)

E Nn=1

I'(nce -I- 1) (sin rrip)x n - i n!

(v) When a> 1, [3

-1, and x -> co,

(14.34) p (1, x) =

1

(_ l)n _i F(nan!

1) (sin irnpa)x , -1 4_

n=1 (vi) Either when a < 1,(14.35) 1P(1,

= 1, s> 0,x -* 0 or when a > 1, = -1, x -> co,n=1 Qn(a*)(a*)n 0(-N-1)]

= 1.72,rcite 2-a)/(2a) e- [1 +

where = 1 - al (x /a)'/(' -1) (if a 1), = e-1 (if a = 1), K = 11(if a 1), K 1 (if a = 1), a* = a A (1/a), Qi(a*) = -11 (2 -I- a* 2a* 2 ), and, 1 in general, Qn (a* ) is a polynomial of degree 2m in the variable a*.

14. REPRESENTATION OF STABLE AND SEMI-STABLE (vii) When a = 1, 0 1, and x (14.36)

89

p (1,x)

1 N Pn(logx) x-n-1 + 0(x -N-2 (log x )N), E n!

where Pi (logz)=

(1 + M and, in general, Pa (logx) = Einfoi rm(logx)i withm sin

rin = 71E1 ()(17)(-1) 7n-10m -1)(n + 1) gm (El +m=1

"(nr) .

Here r(--i) denotes the (m /)th derivative of the r-function. oo, The simplest special cases are that, as x

(14.37)

p (1,x)

lf(a 1)(sin irpa)x -' 1 4Ex -2K A.(2-0)/(20)e-e V2ir a ."

if a $ 1, /3 1, if a = 1, 0 1, if a > 1, = 1.

The following representation in Zolotarev [5361 of strictly stable distributions is sometimes useful. It does not need special treatment of the cases a =1 and 2. THEOREM 14.19. Let 0 < a < 2. If fi is a strictly a-stable distribution

on IR , then(14.38)

Tl(z) = exP

re -10'120 sgii z) ,

where cl > 0 and 0 G JR with101< ) A 1. The parameters e1 and 0 are uniquely determined by ,u. Conversely, for any c1 and 0, there is a strictly a-stable distribution j satisfying (14.38).

REMARK 14.20. Theorem 14.19 includes the case of 6-distributions other than 60 . See Remark 13.17. In the case a 2, the relationship of the two representations of a strictly a-stable distribution in Theorems 14.15 and 14.19 is as follows:

p2 (tan(c2 + 72)1/2 ,

)

2 ) 1 /2 ,

a 1, a = 1, a < 1, a > 1, a = 1, c 0, oz = 1, c = 0,

-7-, arctan()3 tan '-7),0(2U)) (2-a) 2 = -- -77 , arctan(P tan 1r 2 ) ,{

arctan lc , sgn T,

where arctan is the value in (- 1 , i). When 0 increases from 1 to 1, the 72

parameter 0 increases from 1 to 1 if a < 1, and decreases from 2--c.k- to a --ama if 1 1, there is a probabilitymeasure Pb op IRd such that (15.1) fi(z) =-

It is called semi-selfdecomposable if there are some b > 1 and some infinitely divisible probability measure Pb satisfying (15.1). If p is semiselfdecomposable, then b in the definition is called a span of p.

15. SELFDECOMPOSABLE AND SEMI-SELFDECOMPOSABLE

91

EXAMPLE 15.2. Any stable distribution on Rd is selfdecomposable. Any semi-stable distribution on 1R with b as a span is semi-selfdecomposable with b as a span. To prove this, let p be non-trivial and a-stable, as trivial distributions are evidently selfdecomposable. For any a > 0, there is c such that ii(z)a = ii-(a1 /'z)e1(c,z) . Given b> 1, let a = P and notice that

P -(1)-1 z)a = ii(z)ei(4-14 . It follows that satisfies (15.1) with 4(z) = . Hence p is selfdecomposable. Proof for semi-stable distributions is similar. The class of selfdecomposable distributions is comprehended as a class of limit distributions described below. THEOREM 15.3. (i) Let {Z n : n = 1,2, ...} be independent random variables on 110 and Sn = En,_i Zk. Let p be a probability measure on k Rd . Suppose that there are bn > 0 and cn E Rd for n = 1,2, ... such that (15.2) and that (15.3) {b oZk : k = 1, ... ,n; n = 1, 2, ... } is a nvil array.Pbnga-A-c, 4 IL as

n

CO

Then, p is selfdecomposable. (ii) For any selfdecomposable distribution p on Rd we can find {Z n} independent, bn > 0, and en G Rd satisfying (15.2) and (15.3). An analogous characterization of semi-selfdecomposable distributions as limit distributions of a certain kind of subsequences of ISn I is possible (Maejima and Naito [299]). LEMMA 15.4. Suppose that p is non-trivial. If {Z} independent, bn > 0, and cn E Rd satisfy (15.2) and (15.3), then bn 0 and bn.fi lbn --+ 1 as n co. Proof The condition (15.3) says that, for any1 E] * O.

Suppose that some subsequence {b o,} of {N} tends to a non-zero b. Then it follows that, for any k, PEtZkl > b'El= O. Hence Zk = 0 a.s. Therefore p