# Weak Convergence of Measures

## 1st Edition

### Probability and Mathematical Statistics: A Series of Monographs and Textbooks

**Authors:**Harald Bergström

**Editors:**Z. W. Birnbaum E. Lukacs

**eBook ISBN:**9781483191454

**Imprint:**Academic Press

**Published Date:**28th October 1982

**Page Count:**260

## Description

Weak Convergence of Measures provides information pertinent to the fundamental aspects of weak convergence in probability theory. This book covers a variety of topics, including random variables, Hilbert spaces, Gaussian transforms, probability spaces, and random variables.

Organized into six chapters, this book begins with an overview of elementary fundamental notions, including sets, different classes of sets, different topological spaces, and different classes of functions and measures. This text then provides the connection between functionals and measures by providing a detailed introduction of the abstract integral as a bounded, linear functional. Other chapters consider weak convergence of sequences of measures, such as convergence of sequences of bounded, linear functionals. This book discusses as well the weak convergence in the C- and D-spaces, which is reduced to limit problems. The final chapter deals with weak convergence in separable Hilbert spaces.

This book is a valuable resource for mathematicians.

## Table of Contents

Preface

Chapter I Spaces, Mappings, and Measures

1. Classes of Sets

2. Alexandrov Spaces, Topological Spaces, and Measurable Spaces

3. Mappings

4. Classes of Bounded, Real-Valued, Continuous Functions and Measurable Functions

5. Normal Spaces and Completely Normal Spaces

6. Sequences of Sets

7. Metric Spaces

8. Mappings into Metric Spaces

9. Product Spaces

10. Product Spaces of Infinitely Many Factors

11. Some Particular Metric Spaces

12. Measures on an Algebra of Subsets

13. Measures on A-Spaces

14. Extensions of Measures

15. Measures on Infinite-Dimensional Product Spaces

16. Completion of Measures, Continuity Almost Surely and Almost Everywhere

Chapter II Integrals, Bounded, Linear Functionals, and Measures

1. Integrals as Nonnegative, Bounded, Linear Functionals

2. Generalizations of the Abstract Integral

3. The Representations of Bounded, Linear Functionals by Integrals

4. Measures Belonging to a Nonnegative, Bounded, Linear Functional on a Normal A-Space

5. Transformations of Measures and Integrals

6. Constructions of Measures on Metric Spaces by Riemann-Stieltjes Integrals

7. Measures on Product Spaces

8. Convolutions of Measures

9. Probability Spaces and Random Variables

10. Expectations, Conditional Expectations, and Conditional Probabilities

11. The Jensen Inequality

Chapter III Weak Convergence in Normal Spaces

1. Weak Convergence of Sequences of Measures on Normal Spaces

2. Weak Convergence of Sequences of Induced Measures and Transformed Measures

3. Uniformly σ-Smooth Sequences of Measures

4. Weak Limits of σ-Smooth Measures on Completely Normal A-Spaces

5. Reduction of Weak Limit Problems by Transformations

6. The Reduction Procedure for Metric Spaces

7. Weak Convergence of Tight Sequences of Measures on Metric Spaces

8. Seminorms on an Algebra

9. Some Fundamental Identities and Inequalities for Products

10. Convergence in Seminorms of Powers to Infinitely Divisible Elements

11. Convergence in Seminorms of Products

Chapter IV Weak Convergence ON R(k)

1. σ-Smooth Measures on R(k)

2. Gaussian Measures and Gaussian Transforms

3. Fourier Transforms and Their Relation to Gaussian Transforms

4. Gaussian Seminorms

5. The Semigroup of σ-Smooth Measures

6. Stability Conditions for Convolution Products That Converge Weakly

7. The Unique Divisibility of Infinitely Divisible σ-Smooth Measures

8. Lévy Measures on R(k); Gaussian Functionals

9. Weak Convergence of Convolution Powers of σ-Smooth Measures

10. The Semigroup of Infinitely Divisible σ-Smooth Measures

11. The Characteristic Function of an Infinitely Divisible Probability Measure on R{k) and Its Connection with the Gaussian Functional

12. Weak Convergence of Convolution Products

13. Stable Probability Measures

14. Gaussian Transforms and Gaussian Seminorms of Random Variables: A Comparison Method

15. Weak Limits of Distributions of Sums of Martingale Differences

16. Weak Limits of Distributions of Sums of Random Variables under Independence and φ-Mixing

Chapter V Weak Convergence on the C- and D-Spaces

1. The C- and D-Spaces

2. Projections

3. Approximations of Functions by Schauder Sequences

4. Weak Convergence

5. Fluctuations and Weak Convergence

6. Construction of Probability Measures on the C- and D-Spaces

7. Gaussian σ-Smooth Measures on the C- and D-Spaces

8. Embedding of Sums of Real-Valued Random Variables in Random Functions into the D-Space

9. Empirical Distribution Functions

10. Embedding of Sequences of Martingale Differences in Random Functions

Chapter VI Weak Convergence in Separable Hilbert Spaces

1. σ-Smooth Measures on l2-Space

2. Weak Convergence of Convolution Products of Probability Measures on l2

3. Necessary and Sufficient Conditions for the Weak Convergence of Convolution Products of Symmetrical Probability Measures

4. Necessary and Sufficient Conditions for the Weak Convergence of Convolution Powers of Probability Measures

5. Different Forms of Necessary and Sufficient Conditions for the Weak Convergence of Convolution Powers of Probability Measures on l2

6. Invariants of Infinitely Divisible σ-Smooth Measures on l2 Gaussian Functionals

7. The Characteristic Function of Probability Measures on l2

Appendix

A Product-Sum Identity

Notes and Comments

Bibliography

Index

## Details

- No. of pages:
- 260

- Language:
- English

- Copyright:
- © Academic Press 1982

- Published:
- 28th October 1982

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483191454

## About the Author

### Harald Bergström

## About the Editor

### Z. W. Birnbaum

### E. Lukacs

### Affiliations and Expertise

Bowling Green State University