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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.
Sample Chapter(s)
Chapter 1: Probability Measures on Metric Spaces (318 KB)
Contents:
- Probability Measures on Metric Spaces
- The Fourier Transform in a Banach Space
- The Structure of Infinitely Divisible Probability Measures
- Harmonic Analysis of Convolution Semigroups
- Negative Definite Functions and Convolution Semigroups
- Probabilistic Properties of Convolution Semigroups
- Hypergroups in Probability Theory
- Limit Theorems on Locally Compact Abelian Groups
Readership: Graduate students, lecturers and researchers in probability and statistics.
Reviews of the First Edition
“This is an attractive book.”
“This book is well organized and very readable. The list of frequently used symbols and the index are useful. It is made as self-contained and simple as is possible and reasonable.”
“This is a well-written, practically self-contained (and suitable for courses) account of the Lévy-Khinchin representation of infinitely divisible measures and convolution semigroups in two settings.”
“This is an attractive book.”
Short Book Reviews
“This book is well organized and very readable. The list of frequently used symbols and the index are useful. It is made as self-contained and simple as is possible and reasonable.”
Mathematical Reviews
“This is a well-written, practically self-contained (and suitable for courses) account of the Lévy-Khinchin representation of infinitely divisible measures and convolution semigroups in two settings.”
Zentralblatt MATH
Sample Chapter(s)
Chapter 1: Probability Measures on Metric Spaces (318k)
