Stable Probability Measures on Euclidean Spaces and on Locally Compact Groups

Mathematics and Its Applications

Managing Editor:

M. HAZEWINKEL

Centre for Mathematics and Computer Science. Amsterdam. The Netherlands

Volume 531

Stable Probability Measures on Euclidean Spaces and on Locally Compact Groups Structural Properties and Limit Theorems

by

Wilfried Hazod Mathematical Department, University of Dortmund, Germany

and

Eberhard Siebert (t)

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-5832-4 ISBN 978-94-017-3061-7 (eBook) DOI 10.1007/978-94-017-3061-7

Printed on acid-free paper

All Rights Reserved © 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 Softcover reprint of the hardcover 1st edition 2001 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

Preface

The idea of preparing a monograph about stability on groups grew out from a long period of cooperation and discussion between the authors, from the very beginning of their academic careers, both fascinated by probabilities on algebraic structures and by the interplay of harmonic analysis and probability theory. We were both in Tiibingen when H. Heyer's monograph 'Probability Measures on Groups' was written, a time and a subject which had a considerable influence on our future research. And again it was H. Heyer who encouraged us to begin the project of this book: At the beginning of the nineties the investigations in (semi-) stability had achieved a state of maturity which seemed to merit representation in a unified way and from a common point of view.

We developed a detailed plan of what should be included, pointing out which investigations - in particular for groups - should be carried out. During a sabbatical semester in summer 1993, Eberhard Siebert started to prepare the first Chapter. The main part was written when we met at a conference in Linz in September and we used the opportunity to discuss this part of our book.

That was the last time we met. Soon afterwards I was shocked by the message that Eberhard had died on 10th October 1993, which was very sudden and unexpected for his family.

I had lost not only a co-author but also a friend, which deeply saddened me. Nevertheless, it was not long after this that I decided to continue our joint project. However, it turned out that for several years I felt unable to make much progress in restarting the work. Of course, mathematical research is dynamic, and meanwhile new results had been published which should have been covered by the book. Therefore the initial plan had to be adapted several times, but the major part of the book still follows the original project. The final version develops, in Chapter I and II, parallel features on vector spaces and on homogeneous groups, respectively, and presents in Chapter III a survey of the state of knowledge for arbitrary locally compact groups. The core of the last Chapter consists of results to which the authors have contributed in an esential way. However, it turned out to be difficult to collect and rearrange the material, and new detailed investigations were particularly necessary in order to obtain finally - at least for Chapter II - a smooth and more or less complete representation.

iii

iv PREFACE

During the preparation I benefitted from discussions, critical comments and hints from many colleagues, and lowe my gratitude for having been given preprints before publication. I mention, in particular, H. Heyer, Zb. Jurek, Yu. Khokhlov, A. Luczak, D. Neuenschwander, G. Pap, H.-P. Scheffler, R. Shah and Hm. Zeuner. In addition I would like to thank K. Tel6ken and P. Becker-Kern for their critical reading of early versions.

It is a pleasure for me to express my thanks to Dagmar Hennies-Hahn and Frank Hahn. They were among my first students in Dortmund, and, although they fin­ished their studies in the early eighties, they continue their interest in mathematics. Consequently, for their own pleasure, they were kind enough to carefully read the entire manuscript, discovering numerous mistakes and errors. Because of their crit­ical comments the text was improved and became - I hope - more readable, for non-specialists.

Finally, I want to express my thanks to Kluwer Academic Publishers for their patience in waiting such a long time for the manuscript.

When Eberhard Siebert died, the scientific community lost an excellent math­ematician. I lost a friend, but we can hardly imagine what this loss meant to his family. My sympathy is with them. This book is dedicated to Karin Siebert and to the children.

w. Hazod Dortmund, December 1999

Contents

Preface iii

Introduction xi

I Probabilities on vector spaces 1

§ 1.1 Preparations: Linear operators on finite-dimensional vector spaces 3

I Notations (in particular for Chapter I) .... 4

II Discrete one-parameter groups of operators . .

III Continuous one-parameter groups of operators

IV Linear groups ................. .

§ 1.2 Full probability measures and convergence of types § 1.3 Operator-semistable laws and operator-stable laws .

I Definition and Levy-Khinchin representation .

II Annexe: More on infinitely divisible laws . . .

§ 1.4 Levy measures of operator- (semi-) stable laws

I Levy measures of operator-semistable laws . .

II Levy measures of operator-stable laws. . . . .

§ 1.5 Algebraic characterization of operator- (semi-) stability

I The structure of Lin(J.L) . . . . . . . . . . . . . . . .

II Subordination and (semi-) stability . . . . . . . . .

III A randomized characterization of operator-stability

§ 1.6 Operator- (semi-) stable laws as limit distributions

I Domains of operator- (semi-) attraction ..... . II Annexe: More on limits of infinitely divisible laws

III More on domains of operator semi-attraction . § 1. 7 Properties of operator- (semi-) stable laws . . .

§ 1.8 Exponents of operator-stable laws . . . . . . . .

§ 1.9 Elliptical symmetry and large symmetry groups

I Elliptically symmetric operator- (semi-) stable laws II Large symmetry groups . . .

v

6

7

11

11 17 17 25

26 26 29

35

35 41 42

44

44 49 56 62 67 74 74 79

vi CONTENTS

§ 1.10 Domains of normal operator attraction 81 I Stable laws. . . . . . . . . . . . . . . 81 II Remarks on operator-semistable laws . 90

III Moments and domains of attraction . . 92

§ 1.11 The existence of commuting normalizations. 96 § 1.12 More on the structure of the decomposability group Lin(JL) 100

I Semistability and strict semistability ............ 100 II Jordan decomposition and spectrum of normalizing operators 106

III Marginal distributions of operator (semi-) stable laws 109 § 1.13 More on convergence of types theorems . . . . . . 116

I Types and transformation groups . . . . . . . . . 117

II Applications of the convergence of types theorem 120

III Finite-dimensional vector spaces . . . . . . . . 124

IV A method to construct full measures, given B .. 125 V Some examples. . . . . . . . . . . . . . . . . . . . 129

VI Stochastic compactness and regular variation properties . 133 § 1.14 Probabilities with idempotent type. r-stable and completely stable

measures . . . . . . . . . . . . . . . . . . 135 § 1.15 Examples and counterexamples ..... 147

I Operator-stable laws on V = IR 2 and IR 3 147

II Subordination of stable laws . . . . . . . III Probabilities with discrete symmetry group on V = IR 2

IV Marginal distributions of operator stable laws V Convergence of types and idempotent types

VI Limit laws and domains of attraction . . VII Commuting normalizations . . . . . . . .

§ 1.16 References and comments for Chapter I .

150

152

158 160

164 170 171

II Probabilities on simply connected nilpotent Lie groups 181

§ 2.0 Probabilities on locally compact groups: Some fundamental theorems 183

I Continuous convolution semigroups and the structure of generating functionals . . . . . . . . . . . . . . . . . . . . . . . 183

II Convergence of continuous convolution semigroups . 188 III Discrete convolution semigroups . . . . . . . 194 IV Embedding theorems . . . . . . . . . . . . . . . . . 195

V Annexe: Supports of convolution semigroups . . . . 199 § 2.1 Probabilities on simply connected nilpotent Lie groups 200

I Discrete and continuous convolution semigroups: The translation procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

II Automorphisms and contractible Lie groups. Some basic facts . . . 203

CONTENTS vii

III Some examples of contractible Lie groups . 209 § 2.2 Convergence of types and full measures 213

I Simply connected nilpotent Lie groups .. 213 II Some generalizations ............ 220

§ 2.3 Semistable and stable continuous convolution semigroups on simply connected nilpotent Lie groups . . . . . . . . 223

§ 2.4 Levy measures of stable and semistable laws 231 I

II § 2.5

I

II

Levy measures of semistable laws . . . . . . Levy measures of stable laws . . . . . . . . .

Algebraic characterization of (semi-) stability. The structure of Lin(p) ....... .

The structure of Inv(p), resp. Inv(A) ..... III Subordination and semistability . . . . . . . .

IV A randomized characterization of operator-stability § 2.6 (Semi-) stable laws as limit distributions . . . . . .

231 233

235 235 240

245

246 247

I Limit theorems and uniqueness of embedding for semistable laws. 247 II Domains of (semi-) attraction ...

§ 2.7 Properties of (semi-) stable laws . . . .

I Absolute continuity and purity laws . .

256 263

263

II Gaussian and Bochner stable measures 267 III Holomorphic convolution semigroups 271 IV Moments of (semi-) stable laws. . . . . 275

§ 2.8 Exponents of stable laws . . . . . . . . 281

§ 2.9 Elliptical symmetry and large invariance groups 288 § 2.10 Domains of normal attraction ... 294

I Stable and semistable laws . . . . . . . . . . . . 294 II Moments and domains of attraction . . . . . . . 300

§ 2.11 Probabilities with idempotent type: f-stable and completely stable measures . . . . . . . . . . . . . . . . . . . . . . . . . 303

I Idempotent (infinitesimal) f-types and f-stable laws II Complete stability . . . . . . . . . .

303 314

III Marginals and complete stability . . . . . . . . . . . . 322 IV Intrinsic definitions of semistability . . . . . . . . . . 326

§ 2.12 Domains of partial attraction and random limit theorems on groups and vector spaces . . . . . . . . . . . . . . . . 328

I The existence of universal laws (Doeblin laws) . . . 328 II Stochastic compactness . . . . . . . . . . . . . . . . 336 III Random limit theorems: Independent random times 337

§ 2.13 Geometric (semi-) stability 354 I Geometric convolutions . . . . . . . . . . . . . . . . 354

viii CONTENTS

II Properties of geometric and exponential distributions . . . . . . . . 355 III Characterization of geometric convolutions and exponential mixtures 358

IV Geometric semistability . . . . . . . . . . . . . . . . 363

V Geometric domains of attraction . . . . . . . . . . . . 364

VI Illustrations and examples for vector spaces G = V . 366

VII More arithmetic properties of geometric convolutions 368

§ 2.14 Remarks on self-decomposable laws on vector spaces and on groups 371

I The decomposability semigroup D(p,) . . . . . . . . . . . . . . . . . 371 II Self-decomposability........................... 375

III Co cycle equations, background driving processes and generalized Ornstein-Uhlenbeck processes . . . . . . . . 376

IV Stable hemigroups and self-similar processes 379 V Space-time processes ............. 382

VI Processes on G and on V .......... 383

VII Background driving processes with logarithmic moments 384

VIIIFull self-decomposable distributions and limit laws. . . . 390 IX Generalizations and examples .... . . . . . . . . . . . 392

§ 2.15 More limit theorems on G and V: Mixing properties and dependent random times ...................... 398

I A theorem of H. Cramer ........................ 399

II Limit theorems for mixing arrays of random variables . . . . . . . . 402

III Random limit theorems in the domain of attraction of (semi-) stable laws: Dependent random times . . . . . .

§ 2.16 References and comments for Chapter II .....

III (Semi-) stability and limit theorems on general locally groups

§ 3.1 Contractive automorphisms on locally compact groups I Contractive automorphisms and contractible groups

II Totally disconnected contractible groups . . . . .

III The structure theorem for contractible groups

IV Contractive one-parameter automorphism groups

406 413

compact 427 428

429

433

438 440

V Some more structure theorems for discrete automorphism groups 446

§ 3.2 Automorphisms contracting modulo a compact subgroup K ... 448 I Contraction mod K . . . . . . . . . . . . . . . . . . . . . . . . .. 449

II The structure theorem: C K (r) = C (r) . K for discrete automorphism groups acting on a Lie group . . . . . . . . . . . . . . . . . . . . . . 454

III Borel cross-sections for the action of C(r) on CK(r) (discrete auto-morphism groups) . . . . . . . . . 458

IV Continuous automorphism groups . . . . . . . . . . . . . . . . . . . 462

CONTENTS ix

V The structure theorem: CK(T) = C(T) )4 K for continuous auto-morphism groups ........................ 465

VI The structure of C K ( T) for p-adic Lie groups . . . . . . . . . 467 § 3.3 Examples, counterexamples and some more structure theory 468

I Contractible and K-contractible Lie groups. . 468 II Automorphisms of compact groups .............. 475 III Infinite-dimensional tori and solenoidal groups . . . . . . . . 477 IV Retopologization of C(T): Intrinsic topologies of contractible groups 480

§ 3.4 (Semi-) stable convolution semigroups with trivial idempotents . . . 488 I General definitions of strictly (semi-) stable convolution semigroups 488 II (Semi-) stable continuous convolution semigroups with trivial idem-

potents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 III Some examples and further remarks . . . . . . . . . . . . . . . . . . 498

§ 3.5 (Semi-) stable convolution semigroups with nontrivial idempotents. 507 I Semistable convolution semigroups on Lie groups with nontrivial

idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . 509 II Stable convolution semigroups with nontrivial idempotents . . . . . 512 III Semistable sUbmonogeneous semigroups on Lie groups. . . . . . . . 514 IV Semistable convolution semigroups with nontrivial idempotents on

p-adic Lie groups .................... 518 § 3.6 More on probabilities on contractible groups . . . . .

I Domains of partial attraction on contractible groups .

519 519

II The existence of Doeblin laws on contractible groups 528 III A translation procedure for contractible locally compact groups 530 IV Point processes on groups and continuous convolution semigroups 534

§ 3.7 Limit laws and convergence of types theorems. A survey ..... 537 I Limits of discrete convolution semigroups with nontrivial idempotents537 II Convergence of types theorems . . . . . . . . . . . . . . . . . . . 549 III Applications to semistability . . . . . . . . . . . . . . . . . . . . 557 IV Limit laws on compact extensions of contractible groups N )4 K 559

§ 3.8 References and comments for Chapter III . . . . . . . . . . . . . 562

Epilogue 567

Bibliography 573

List of Symbols 601

Index 607

Introduction

The starting points for investigating stable probabilities on the real line were the investigations of A. Khinchin and P. Levy into the limit behavior of normalized sums of independent identically distributed real random variables (cf. [257] (1925), [219] (1933), [222] (1936)). The class of stable limit laws was characterized in different ways:

1) The characteristic functions fulfil a functional equation (similar to Gaussian laws) "ji(ta. x) = "jit(x) for some a > 0, for all x E JR, t > 0; whence:

2) the embedding continuous convolution semigroup (l1,t)t'2 0 is generated by 'space­transformations', p,t = HtQ (p,), where H y denotes the homothetic transforma­tion Hy(x) := y . x;

3) for all a, b E JR x there exists acE JR x such that "ji(a . x) . "ji(b . x) = "ji(c . x); equivalently, Ha(P,) * Hb(p,) = Hc(p,). In other words, if we define type(p,) := {Hy(p,) : y :j:. O}, then 3) is equivalent to idempotence of the type: type(p,) * type(p,) = type(p,); finally:

4) nondegenerate stable laws are characterized as limits of distributions of normal­ized sums of sequences of LLd. random variables an' E~ Xi = Han (E~ Xi), i.e., as probabilities with nonvoid domains of attractions.

Property 3) is used as the definition of stability, and the equivalence of 1)-4) has been shown. In fact, these laws were originally called stable (cf., [257] (1925), [259]) and are nowadays usually called strictly stable. If, instead of homothetic normalizations, affine transformations are allowed, a larger class of laws is obtained - these laws were called quasistable (cf. [258] (1935), [259], [222]), and are nowadays usually called stable, cf., e.g., V. Gnedenko and A.N. Kolmogorov [93] (1954). A comprehensive description of stable laws on JR can be found in the monograph by V.M. Zolotarev [451] (1986).

The class of (quasi-) stable laws - defined by 3) - was generalized into different directions: P. Levy [258], [259] introduced the class of semistable and quasi-semistable laws, defined as laws fulfilling more general functional equations "ji( a . x) = "jib (x) . ei .cx for suitable a, b > 0, c E JR, and all x E JR. On the other hand, P. Levy introduced the class L ('lois limites') as limits of distributions of normalized sums an . E~ Xi - bn , where Xi are independent (not necessarily identically distributed) and {anXih~n,nEN is an infinitesimal array, cf., [259] (1937, 1954); see also the monograph [93] (1954)

xi

xii INTRODUCTION

or the monograph by M. Loeve [265]. Nowadays these laws are frequently called sel/­decomposable or Levy measures.

The characterization of semistable laws as limit laws, analogous to 4), seems not to have been published at the beginning of these investigations, although we may assume that the existence of such relations was known. This is indicated by the reflections on domains of (partial) attraction of semistable laws in [259, Chap. VII, 57-60], and by remarks in W. Doeblin's thesis [59] (1940). However, no precise statements nor proofs seem to have been published. The first proofs were published independently by V. Kruglov [244] (1972), D. Mejzler [312] (1974) (without referring to P. Levy's definition), R.N. Pillai [347] (1971), and R. Shimizu [390], [389] (1969,1970), including also complete descriptions of the Levy-Khinchin representation of semistable laws.

In fact, the above-mentioned probabilities form a subset of the infinitely divisible laws. This class of laws was characterized by A. Khinchin and W. Doeblin, cf., e.g., [221] (1937), [59] (1940), as limits of distributions of normalized sums of sequences of Li.d. random variables an· L~n Xi, i.e. laws with nonvoid domains of partial attraction.

Multivariate generalizations of (semi-) stability with homothetic normalizations appeared immediately, beginning, e.g., with [74], see also [259], and developed in vari­ous directions; the reader will find an overview (with emphasis on infinite-dimensional spaces) in W. Linde's monograph [261] (1986); see also K. Sato's more recent mono­graph [368].

The investigation of operator-normalized limit distributions on vector spaces started at the same time as the investigations into semistability, thirty years after the initia­tion by P. Levy and A. Khinchin [222], [259], with the pioneering works of M. Sharpe [387] (1969; operator-stable laws), R. Jajte [188] (1977, the operator-semistable case), and K. Urbanik [429] (1972, operator Levy measures). (In fact, the first investiga­tions into operator-stability are found in G.N. Sakovic's thesis [365] (1965). But these investigations remained unnoticed for long time. Cf. e.g. [213].)

The description of operator-stability by the analog of property 2) appears in the following form: A probability p, E Ml (V) on a vector space V, embedded into a continuous convolution semigroup (p,t)t>o, is (strictly) operator-stable if there exists a continuous group (tE = exp((lnt)· E))t>o ~ GL(V) such that p,t = tE(p,), t > O. Operator-semistability is defined analogously. The equivalence of 1), resp. 2) and 4), the characterization of full operator- (semi-) stable laws as limit laws, is proved in [387], resp. [188]. Moreover, operator-stability is again equivalent to 3) - if the type is defined as {tE(p,) Hz: t > O,x E V}.

In parallel to this, more or less concurrently, a further milestone was published: U. Grenander's short and ingenious monograph 'Probabilities on Algebraic Struc­tures' [103] (1968), suggesting a program for further investigations, especially into probabilities on vector spaces and on locally compact groups. This program already mentions the importance of investigating limits of sums, resp. products, of i.i.d. ran­dom variables normalized by suitable transformations ([103], outlook), anticipating the above-mentioned operator-normalizations on vector spaces and normalizations by

xiii

automorphisms on groups. Simultaneously, the first systematic treatement in book form of probabilities,

and in particular of limit laws, on Abelian groups appeared with the monograph of K.R. Parthasarathy [344] (1967).

A part ofU. Grenander's program for locally compact groups was completed in the nineteen-seventies; the state of the art was published in H. Heyer's monograph [153] (1977). There the reader finds a systematic and comprehensive treatment of proba­bilities on locally compact groups and of the probabilistic relevance of certain classes of locally compact groups; in particular of infinitely divisible and of continuously em­beddable laws, of continuous convolution semigroups and their generating functionals (Levy-Khinchin representation), and limit theorems for triangular systems on locally compact - not necessarily Abelian - groups.

However, the limit behavior of automorphism-normalized Li.d. random variables, in particular (semi-) stable laws on groups, are not yet considered there. Before start­ing a systematic investigation in this direction it was necessary to have beforehand:

• sufficient examples of groups admitting contractive automorphisms, in order to generate infinitesimal arrays of operator-normalized Li.d. random variables;

• to describe the stucture of such contractible groups; and • to realize that limit laws of automorphism-normalized products are concentrated

on contractible subgroups.

Equipped with these tools it was possible to start with investigations of (semi-) stability on groups - defined by a generaliziation of the above-mentioned functional equation 2) analogous to the definitions of operator- (semi-) stability on vector spaces. (Since Fourier transforms are not available, property 1), resp. the vector space version, has to be replaced by the equivalent version 2) for convolution semigroups, resp. for generating functionals.)

One essential point for the further development is the fact that homogeneous groups, i.e., Lie groups admitting dilations, a group of contractive automorphisms, became an object of interest from the point of view of harmonic analysis. See, e.g., R. Goodman [99] (1977), [100], [98]; see also P.R. Muller-Romer [323] (1976), F.G. Fol­land [79] (1989) ff., and the bibliographical hints mentioned there. A complete char­acterization of contractible locally compact groups was published in E. Siebert [409] (1986), showing in particular that contractible connected groups are just homogeneous groups and these are again characterized as simply connected nilpotent Lie groups with positively graduated Lie algebra.

Next, (dilation-) stable continuous convolution semigroups on homogeneous groups became an object of investigation in harmonic analysis (see e.g. A. Hulanicki [181]­(183), P. Glowacki [86)-[90)), as well as from the probabilistic point of view. The latter is documented by the proceedings of the Oberwolfach conferences on 'Probabilities on Groups' (1978-1994), e.g., P. Baldi [4], Th. Drisch, L. Gallardo [63], E. Siebert [403), [410], resp. [119], [120], and the survey in [123], where the structure and Levy­Khinchin representation of (semi-) stable laws is discussed. A representation of stable

xiv INTRODUCTION

laws as limit laws as in the operator- (semi-) stable case was obtained in [4] and [63] mentioned above; for general contractible Lie groups the characterization of (semi-) stability by limit laws is contained in S. Nobel's thesis [334] (1989), [335], based on a suitable convergence of types theorem, cf. [141], which turned out to be crucial.

Idempotence of types and r-stable laws,generalizing property 3), appear for vector spaces in the investigations of K.R. Parthasarathy and K. Schmidt [346], [345], [379] (1975), followed by the investigations of B. Mincer and K. Urbanik [316]-[318] (1982); for a group version see [127], [129].

A series of new contributions to operator- (semi-) stability started ten years af­ter M. Sharpe's pioneer work with investigations of, in particular, W.N. Hudson, J.P. Holmes, Zb. Jurek, W. Krakowiak, A. Luczak, J.D. Mason, M.M. Meerschaert, H-P. Scheffler, E. Siebert, H.G. Tucker, J.A. Veeh, and C. Weiner, completing our knowledge of the vector space situation.

Since the publication of the above-mentioned fundamental treatment of limit laws, [93] (1954), stable laws are considered as a special subset of self-decomposable laws, and hence investigations into semistable laws are not included. Modern investiga­tions for vector spaces, in particular the monograph by Zb. Jurek and D. Mason [213] (1993), follow this set-up. In fact, this shows one of the basic differences be­tween the recent investigations into operator-normalized random variables on vector spaces and automorphism-normalized random variables on groups: During the last few decades the behavior of limits of products of LLd. random variables on groups is quite well understood, while for limits of products of infinitesimal non-identically distributed random variables our knowledge is still insufficient. (Although recently important progress was achieved, see, e.g., H. Heyer and G. Pap [158], [159].) Hence the representation of (semi-) stable laws as limit distributions is now quite satisfac­tory - at least for contractible Lie groups - while the results available for self­decomposability on groups are still fragmentary. Therefore, self-decomposability is not included in Chapter I, and only a first overview will be given in Chapter II, show­ing that self-decomposability is closely related to continuous convolution semigroups on a semidirectly extended group.

A further milestone in the development was an observation of Yu. Khokhlov [223] (1991), who discovered that E. Siebert's fundamental paper [398] already contains a result which allows one to compare the limit behavior of group-valued random variables with random variables on the tangent space, a finite-dimensional vector space in the case of Lie groups. In other words, Yu. Khokhlov discovered the missing link between investigations into limit theorems on vector spaces and on (contractible Lie) groups. Hence we obtain a complete correspondence between (semi-) stable laws and their domains of attraction on vector spaces and on (contractible Lie) groups, respectively. This yields a 1-1 correspondence between objects and limit behavior on groups and on vector spaces (cf. §2.1.3-§2.1.7), called a translation procedure and allowing one to extend known limit laws from vector spaces to (contractible Lie) groups and to reduce investigations for groups to the vector space situation.

xv

This monograph follows to some extent the historical development described above and is therefore divided into three parts: Chapter I is devoted to operator-normalization and limit laws on vector spaces, Chapter II to contractible (hence simply connected nilpotent) Lie groups, and in Chapter III we provide a survey of (semi-) stability on general locally compact groups. This survey shows that in the situation of general locally compact groups, in many situations the investigations may also be reduced to Chapters I and II, and there are further large classes of groups, e.g., contractible locally compact groups, where analogous correspondences between semistability and limit laws can be proved. However Chapter III will also show that numerous questions and open problems are awaiting investigation.

The theory for vector space-valued random variables is, at the moment, rapidly developing and promises interesting applications. In comparison, the theory for group­valued random variables is still less elaboratedj nevertheless, at least for simply con­nected nilpotent Lie groups we have a more or less complete overview, and for a large part of the theory we obtain 1-1 correspondences between corresponding investiga­tions on groups and on vector spaces respectively by the above-mentioned translation procedure. The main aim of this book is therefore to point out these parallel fea­tures on vector spaces and on groups. Thus, in particular, the material chosen in Chapter I is limited: Interesting investigations on vector spaces which have as yet no corresponding theory for groups are omitted or only marginally treated. On the other hand, investigations on groups which have no corresponding theory on vector spaces or need more profound tools from analysis or harmonic analysis, e.g., pseudo-differential operators or structure of algebraic groups, are therefore also omitted.

A guide for the reader

This book is written to serve as an introduction to the above-mentioned topics for researchers and graduate students and - presenting a large part of the investigations of the last thirty years from a common point of view - also as a reference text for specialists interested only in particular topics. There exist only a few treatises on closely related subjects. As already mentioned, the monograph by Zb. Jurek and D. Mason [213] is concerned with vector spaces and D. Neuenschwander's treatise [327] concentrated on Heisenberg groupSj there is also - in preparation - a monograph of Yu. Khokhlov [228]. Of course, the intersection of the present book with any of the above-mentioned monographs is nonvoid but the emphases are different and in the main parts complementary subjects are treated.

The reader mainly interested in the vector space set-up should start with Chap­ter I. However, some parts of the theory which are not of central interest are found in Chapter II, §2.12-§2.15 (developed simultaneously for vector spaces and simply connected nilpotent Lie groups). Thus, a reader whose main interest is in the vector space set-up, in particular in domains of partial attraction, universal laws and in limit laws with random indices (transfer theorems), is referred to §2.12j for geometric con­volutions and geometric (semi-) stability see §2.13, for operator self-decomposability

xvi INTRODUCTION

and related topics see §2.14j for limit laws under mixing conditions and dependent random times see §2.15. This reader should read Chapter I, skip (most of) §2.0-§2.11 and restart with §2.12 ff.

Of course, as mentioned before, self-decomposability appears in §2.14 only in a marginal role, since there do not exist sufficient corresponding investigations in the group set-up. The reader particularly interested in this topic will find more (and more profound) investigations in the literature, e.g., in the monograph by Zb. Jurek and D. Mason [213).

The reader mainly interested in probabilities on groups may consider Chapter I as a motivating introduction with a collection of examples (cf. §1.15) and will begin reading at Chapter II, §2.0. There, beginning with §2.1, we have systematically collected results for simply connected nilpotent Lie groups that have a counterpart on vector spaces. The main part of Chapter II is organized in such a way that the reader will easily find the corresponding results for groups in Chapter II and vector spaces in Chapter I, respectively. Owing to the 'translation procedure' (cf. §2.1.3-§2.1.7), it frequently turns out that the proofs for the group versions are almost verbatim repetitions of the vector space versionsj therefore they are omitted.

It should be mentioned that for particular homogeneous groups, especially for Heisenberg groups, considerably more profound investigations can be found and more particular results are known. The reader is referred to D. Neuenschwander [327), which presents a systematic investigation of possible limit laws for this particular class of groups. For dilation-stability on homogeneous groups and analytical properties of densities, see, e.g., A. Hulanicki and P. Glowacki [181)-[183), [86)-[90], and for more results on central limit theorems on homogeneous groups the reader is referred to, e.g., G. Pap [338)-[341) and the literature mentioned there.

Some parts of the theory presented in Chapter II can also be proved for nilpo­tent p-adic matrix groups. We will only indicate a few such results, in Chapter III (avoiding details from the theory of algebraic groups). The reader interested in re­cent investigations into (semi-) stability on algebraic groups is referred, e.g., to the investigations of S.G. Dani and R. Shah [51), [52), [381)-[386).

Thus, the parts of the theory contained in Chapters I and II do not at all exhaust all (semi-) stability phenomena on groups: In particular, there is an abundance of totally disconnected locally compact groups supporting semistable laws. The reader will find these investigations in Chapter III. There we start with contractible and K-contractible parts of locally compact groups, which will turn out (cf. §3.4 ff.) to support (semi-) stable probabilities. Contractible locally compact groups have a pleas­ant structure, which will be investigated in §3.1-§3.3. (In fact, §3.3 contains examples illustrating the previous structure theory.)

In view of the structure theory developed for contractible and K -contractible subgroups, it turns out that for stable laws - and for semistable laws on Lie groups -the investigations may be reduced to the objects studied in Chapter II (and Chapter I): homogeneous groups (Simply connected nilpotent groups) and vector spaces.

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The theory for vector spaces is now well established. Therefore it was possible to illustrate the theory by a set of examples, which are collected in §1.15. In Chapter II we have no systematic collection of examples, these are spread out in the text, in particular in §2.1 and §2.11. In Chapter III, however, we collected in §3.3 examples illustrating the structure of the groups under consideration; probabilistic examples are again spread throughout §3.4-§3.7.

The investigations of Chapter III end with §3.6 and §3.7, giving an overview of probabilities on contractible groups; in particular, domains of partial attraction and universal laws, functional limit theorems, convergence of types theorems and further possible investigations. This overview will point out connections between (semi-) sta­bility and limit laws in the general set-up and will show which parts of the theory for general groups have only just begun to be investigated and await further future development.

Each chapter is provided with 'References and comments', §1.16, §2.16, resp. §3.8, including also hints and suggestions for investigations that are beyond the framework of this book and not treated here.