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.
Author(s): Herbert Heyer
Series: Series on Multivariate Analysis
Publisher: World Scientific Publishing Co Pte Ltd
Language: English
Pages: 399
Structural Aspects in the Theory of Probability: A Primer in Probabilities on Algebraic - Topological Structures......Page 4
Preface......Page 6
Contents......Page 10
1.1 Tight measures......Page 12
1.2 The topology of weak convergence......Page 17
1.3 The Prohorov theorem......Page 33
1.4 Convolution of measures......Page 39
2.1 Fourier transforms of probability measures......Page 45
2.2 Shift compact sets of probability measures......Page 57
2.3 Infinitely divisible and embeddable measures......Page 71
2.4 Gauss and Poisson measures......Page 79
3.1 The Ito-Nishio theorem......Page 97
3.2 Fourier expansion and construction of Brownian motion......Page 115
3.3 Symmetric Levy measures and generalized Poisson measures......Page 129
3.4 The Levy-Khintchine decomposition......Page 149
4.1 Convolution of Radon measures......Page 172
4.2 Duality of locally compact Abelian groups......Page 186
4.3 Positive definite functions......Page 209
4.4 Positive definite measures......Page 220
5.1 Negative definite functions......Page 238
5.2 Convolution semigroups and resolvents......Page 247
5.3 Levy functions......Page 263
5.4 The Levy-Khintchine representation......Page 272
6.1 Transient convolution semigroups......Page 285
6.2 The transience criterion......Page 300
6.3 Recurrent random walks......Page 320
6.4 Classification of transient random walks......Page 344
A Topological groups......Page 366
B Topological vector spaces......Page 369
C Commutative Banach algebras......Page 377
Selected References......Page 382
Symbols......Page 388
Index......Page 394
