Structural aspects in the theory of probability : a primer in probabilities on algebraic-topological structures / Herbert Heyer.

Format:

Book

Author/Creator:

Heyer, Herbert.

Contributor:

Anne and Joseph Trachtman Memorial Book Fund.

Series:

Series on multivariate analysis ; v. 7.

Series on multivariate analysis ; v. 7

Language:

English

Subjects (All):

Probabilities.

Topological groups.

Banach spaces.

Probability measures.

Abelian groups.

Physical Description:

x, 388 pages ; 24 cm.

Place of Publication:

[River Edge, N.J.] : World Scientific, [2004]

Summary:

Heyer (mathematics, University of Tubingen, Germany) focuses on the algebraic-topological aspects of probability theory, leading to a wider and deeper understanding of basic theorems, such as those related to 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. Graduate students, lecturers, and researchers can use the book as a primer on the theory of probability measures in groups and related structures. Annotation ©2004 Book News, Inc., Portland, OR (booknews.com)

Contents:

1 Probability Measures on Metric Spaces

1.1 Tight measures 1

1.2 The topology of weak convergence 6

1.3 The Prohorov theorem 22

1.4 Convolution of measures 28

2 The Fourier Transform in a Banach Space

2.1 Fourier transforms of probability measures 34

2.2 Shift compact sets of probability measures 46

2.3 Infinitely divisible and embeddable measures 60

2.4 Gauss and Poisson measures 68

3 The Structure of Infinitely Divisible Probability Measures

3.1 The Ito-Nishio theorem 86

3.2 Fourier expansion and construction of Brownian motion 104

3.3 Symmetric Levy measures and generalized Poisson measures 118

3.4 The Levy-Khintchine decomposition 138

4 Harmonic Analysis of Convolution Semigroups

4.1 Convolution of Radon measures 161

4.2 Duality of locally compact Abelian groups 175

4.3 Positive definite functions 198

4.4 Positive definite measures 209

5 Negative Definite Functions and Convolution Semigroups

5.1 Negative definite functions 227

5.2 Convolution semigroups and resolvents 236

5.3 Levy functions 252

5.4 The Levy-Khintchine representation 261

6 Probabilistic Properties of Convolution Semigroups

6.1 Transient convolution semigroups 274

6.2 The transience criterion 289

6.3 Recurrent random walks 309

6.4 Classification of transient random walks 333

A Topological groups 355

B Topological vector spaces 358

C Commutative Banach algebras 366.

Notes:

Includes bibliographical references (pages 371-375) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Anne and Joseph Trachtman Memorial Book Fund.

ISBN:

9812389377

OCLC:

55149283