Stochastic Convergence

Stochastic Convergence, Second Edition covers the theoretical aspects of random power series dealing with convergence problems.

This edition contains eight chapters and starts with an introduction to the basic concepts of stochastic convergence. The succeeding chapters deal with infinite sequences of random variables and their convergences, as well as the consideration of certain sets of random variables as a space. These topics are followed by discussions of the infinite series of random variables, specifically the lemmas of Borel-Cantelli and the zero-one laws. Other chapters evaluate the power series whose coefficients are random variables, the stochastic integrals and derivatives, and the characteristics of the normal distribution of infinite sums of random variables. The last chapter discusses the characterization of the Wiener process and of stable processes.

This book will prove useful to mathematicians and advance mathematics students.

Preface to the Second Edition

Preface to the First Edition

List of Examples

Chapter I. Introduction

1.1. Survey of Basic Concepts

1.2. Certain Inequalities

1.3. Characteristic Functions

1.4. Independence

1.5. Monotone Classes of Sets (Events)

Exercises

Chapter II. Stochastic Convergence Concepts and their Properties

2.1. Definitions

2.2. Relations Among the Various Convergence Concepts

2.3. Convergence of Sequences of Mean Values and of Certain Functions of Random Variables

2.4. Criteria for Stochastic Convergence

2.5. Further Modes of Stochastic Convergence

2.6. Information Convergence

Exercises

Chapter III. Spaces of Random Variables

3.1. Convergence in Probability

3.2. Almost Certain Convergence

3.3. The Spaces Lp

3.4. The Space of Distribution Functions

Exercises

Chapter IV. Infinite Series of Random Variables and Related Topics

4.1. The Lemmas of Borel-Cantelli and the Zero-One Laws

4.2. Convergence of Series

4.3. Some Limit Theorems

Exercises

Chapter V. Random Power Series

5.1. Definition and Convergence of Random Power Series

5.2. The Radius of Convergence of a Random Power Series

5.3. Random Power Series with Identically Distributed Coefficients

5.4. Random Power Series with Independent Coefficients

5.5. The Analytic Continuation of Random Power Series

5.6. Random Entire Functions

Exercises

Chapter VI. Stochastic Integrals and Derivatives

6.1. Some Definitions Concerning Stochastic Processes

6.2. Definition and Existence of Stochastic Integrals

6.3. L2-Continuity and Differentiation of Stochastic Processes

Exercises

Chapter VII. Characterization of the Normal Distribution by Properties of Infinite Sums of Random Variables

7.1. Identically Distributed Linear Forms

7.2. A Linear Form and a Monomial Having the Same Distribution

7.3. Independently Distributed Infinite Sums

Exercises

Chapter VIII. Characterization of Some Stochastic Processes

8.1. Independence and a Regression Property of Two Stochastic Integrals

8.2. Identically Distributed Stochastic Integrals

8.3. Identity of the Distribution of a Stochastic Integral and the Increment of a Process

8.4. Characterization of Stable Processes

Exercises

References

Index