Structural Aspects In The Theory Of Probability (2nd Enlarged Edition)

by Herbert Heyer

Overview

The book is conceived as a text accompanying the traditional graduate courses on probability theory. An important feature of this enlarged version is the emphasis on algebraic-topological aspects leading to a wider and deeper understanding of basic theorems such as those on the structure of continuous convolution semigroups and the corresponding processes with independent increments. Fourier transformation — the method applied within the settings of Banach spaces, locally compact Abelian groups and commutative hypergroups — is given an in-depth discussion. This powerful analytic tool along with the relevant facts of harmonic analysis make it possible to study certain properties of stochastic processes in dependence of the algebraic-topological structure of their state spaces. In extension of the first edition, the new edition contains chapters on the probability theory of generalized convolution structures such as polynomial and Sturm-Liouville hypergroups, and on the central limit problem for groups such as tori, p-adic groups and solenoids.

Product Details

ISBN-13:	9789814282482
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	09/04/2009
Series:	Series On Multivariate Analysis , #8
Edition description:	Enlarged
Pages:	424
Product dimensions:	6.20(w) x 9.00(h) x 1.10(d)

Table of Contents

Preface to the second enlarged edition v

Preface vii

1 Probability Measures on Metric Spaces 1

1.1 Tight measures 1

1.2 The topology of weak convergence 5

1.3 The Prokhorov theorem I8

1.4 Convolution of measures 23

2 The Fourier Transform in a Banach Space 29

2.1 Fourier transforms of probability measures 29

2.2 Shift compact sets of probability measures 39

2.3 Infinitely divisible and embeddable measures 50

2.4 Gauss and Poisson measures 56

3 The Structure of Infinitely Divisible Probability Measures 71

3.1 The Ito-Nisio theorem 71

3.2 Fourier expansion and construction of Brownian motion 86

3.3 Symmetric Lévy measures and generalized Poisson measures 98

3.4 The Lévy-Khinchin decomposition 114

4 Harmonic Analysis of Convolution Semigroups 133

4.1 Convolution of Radon measures 133

4.2 Duality of locally compact Abelian groups 144

4.3 Positive definite functions 162

4.4 Positive definite measures 171

5 Negative Definite Functions and Convolution Semigroups 185

5.1 Negative definite functions 185

5.2 Convolution semigroups and resolvents 191

5.3 Lévy functions 204

5.4 The Lévy-Khinchin representation 211

6 Probabilistic Properties of Convolution Semigroups 225

6.1 Transient convolution semigroups 225

6.2 The transience criterion 237

6.3 Recurrent random walks 253

6.4 Classification of transient random walks 272

7 Hypergroups in Probability Theory 291

7.1 Commutative hypergroups 291

7.2 Decomposition of convolution semigroups of measures 303

7.3 Random walks in hypergroups 318

7.4 Increment processes and convolution semigroups 333

8 Limit Theorems on Locally Compact Abelian Groups 345

8.1Limit problems and parametrization of weakly infinitely divisible measures 345

8.2 Gaiser's limit theorem 349

8.3 Limit theorems for symmetric arrays and Bernoulli arrays 360

8.4 Limit theorems for special locally compact Abelian groups 366

Appendices 375

A Topological groups 375

B Topological vector spaces 377

C Commutative Banach algebras 383

Selected References 389

Symbols 397

Index 403