Structural aspects in the theory of probability: A primer in probabilities on algebraic-topological structures

Herbert Heyer9812389377, 9789812389374, 9789812562289

This book focuses on the algebraic-topological aspects of probability theory, 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. The method applied within the setting of Banach spaces and of locally compact Abelian groups is that of the Fourier transform. This analytic tool along with the relevant parts of harmonic analysis makes it possible to study certain properties of stochastic processes in dependence of the algebraic-topological structure of their state spaces. Graduate students, lecturers and researchers may use the book as a primer in the theory of probability measures on groups and related structures.

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Table of contents :
Cover……Page 1
Series……Page 2
Title page……Page 3
Date-line……Page 4
Preface……Page 5
Contents……Page 9
1.1 Tight measures……Page 11
1.2 The topology of weak convergence……Page 16
1.3 The Prohorov theorem……Page 32
1.4 Convolution of measures……Page 38
2.1 Fourier transforms of probability measures……Page 44
2.2 Shift compact sets of probability measures……Page 56
2.3 Infinitely divisible and embeddable measures……Page 70
2.4 Gauss and Poisson measures……Page 78
3.1 The Ito-Nishio theorem……Page 96
3.2 Fourier expansion and construction of Brownian motion……Page 114
3.3 Symmetric Levy measures and generalized Poisson measures……Page 128
3.4 The Levy-Khintchine decomposition……Page 148
4.1 Convolution of Radon measures……Page 171
4.2 Duality of locally compact Abelian groups……Page 185
4.3 Positive definite functions……Page 208
4.4 Positive definite measures……Page 219
5.1 Negative definite functions……Page 237
5.2 Convolution semigroups and resolvents……Page 246
5.3 Levy functions……Page 262
5.4 The Levy-Khintchine representation……Page 271
6.1 Transient convolution semigroups……Page 284
6.2 The transience criterion……Page 299
6.3 Recurrent random walks……Page 319
6.4 Classification of transient random walks……Page 343
A Topological groups……Page 365
B Topological vector spaces……Page 368
C Commutative Banach algebras……Page 376
Selected References……Page 381
Symbols……Page 387
Index……Page 393