# Fourier Analysis in Probability Theory

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Fourier Analysis in Probability Theory provides useful results from the theories of Fourier series, Fourier transforms, Laplace transforms, and other related studies. This 14-chapter work highlights the clarification of the interactions and analogies among these theories. Chapters 1 to 8 present the elements of classical Fourier analysis, in the context of their applications to probability theory. Chapters 9 to 14 are devoted to basic results from the theory of characteristic functions of probability distributors, the convergence of distribution functions in terms of characteristic functions, and series of independent random variables. This book will be of value to mathematicians, engineers, teachers, and students.

## Table of Contents

Preface

I Introduction

1.1 Measurable Space; Probability Space

1.2 Measurable Functions; Random Variables

1.3 Product Space

1.4 Integrals

1.5 The Fubini-Tonelli Theorem

1.6 Integrals on R1

1.7 Functions of Bounded Variation

1.8 Signed Measure; Decomposition Theorems

1.9 The Lebesgue Integral on R1

1.10 Inequalities

1.11 Convex Functions

1.12 Analytic Functions

1.13 Jensen's and Carleman's Theorems

1.14 Analytic Continuation

1.15 Maximum Modulus Theorem and Theorems of Phragmén-Lindelöf

1.16 Inner Product Space

II Fourier Series and Fourier Transforms

2.1 The Riemann-Lebesgue Lemma

2.2 Fourier Series

2.3 The Fourier Transform of a Function in L1(—∞, ∞)

2.4 Magnitude of Fourier Coefficients; the Continuity Modulus

2.5 More About the Magnitude of Fourier Coefficients

2.6 Some Elementary Lemmas

2.7 Continuity and Magnitude of Fourier Transforms

2.8 Operations on Fourier Series

2.9 Operations on Fourier Transforms

2.10 Completeness of Trigonometric Functions

2.11 Unicity Theorem for Fourier Transforms

2.12 Fourier Series and Fourier Transform of Convolutions

Notes

III Fourier-Stieltjes Coefficients, Fourier-Stieltjes Transforms and Characteristic Functions

3.1 Monotone Functions and Distribution Functions

3.2 Fourier-Stieltjes Series

3.3 Average of Fourier-Stieltjes Coefficients

3.4 Unicity Theorem for Fourier-Stieltjes Coefficients

3.5 Fourier-Stieltjes Transform and Characteristic Function

3.6 Periodic Characteristic Functions

3.7 Some Inequality Relations for Characteristic Functions

3.8 Average of a Characteristic Function

3.9 Convolution of Nondecreasing Functions

3.10 The Fourier-Stieltjes Transform of a Convolution and the Bernoulli Convolution

Notes

IV Convergence and Summability Theorems

4.1 Convergence of Fourier Series

4.2 Convergence of Fourier-Stieltjes Series

4.3 Fourier's Integral Theorems; Inversion Formulas for Fourier Transforms

4.4 Inversion Formula for Fourier-Stieltjes Transforms

4.5 Summability

4.6 (C,1)-Summability for Fourier Series

4.7 Abel-Summability for Fourier Series

4.8 Summability Theorems for Fourier Transforms

4.9 Determination of the Absolutely Continuous Component of a Nondecreasing Function

4.10 Fourier Series and Approximate Fourier Series of a Fourier-Stieltjes Transform

4.11 Some Examples, Using Fourier Transforms

Notes

V General Convergence Theorems

5.1 Nature of the Problems

5.2 Some General Convergence Theorems I

5.3 Some General Convergence Theorems II

5.4 General Convergence Theorems for the Stieltjes Integral

5.5 Wiener's Formula

5.6 Applications of General Convergence Theorems to the Estimates of a Distribution Function

Notes

VI L2-Theory of Fourier Series and Fourier Transforms

6.1 Fourier Series in an Inner Product Space

6.2 Fourier Transform of a Function in L2(-∞, ∞)

6.3 The Class H2 of Analytic Functions

6.4 A Theorem of Szegö and Smirnov

6.5 The Class ƃ2 of Analytic Functions

6.6 A Theorem of Paley and Wiener

Notes

VII Laplace and Mellin Transforms

7.1 The Laplace Transform

7.2 The Convergence Abscissa

7.3 Analyticity of a Laplace-Stieltjes Transform

7.4 Inversion Formulas for Laplace Transforms

7.5 The Laplace Transform of a Convolution

7.6 Operations of Laplace Transforms and Some Examples

7.7 The Bilateral Laplace-Stieltjes Transform

7.8 Mellin-Stieltjes Transforms

7.9 The Mellin Transform

Notes

VIII More Theorems on Fourier and Laplace Transforms

8.1 A Theorem of Hardy

8.2 A Theorem of Paley and Wiener on Exponential Entire Functions

8.3 Theorems of Ingham and Levinson

8.4 Singularities of Laplace Transforms

8.5 Abelian Theorems for Laplace Transforms

8.6 Tauberian Theorems

8.7 Multiple Fourier Series and Transforms

8.8 Nondecreasing Functions and Distribution Functions in Rm

8.9 The Multiple Fourier-Stieltjes Transform

Notes

IX Convergence of Distribution Functions and Characteristic Functions

9.1 Helly Theorems and Convergence of Nondecreasing Functions

9.2 Convergence of Distribution Functions with Bounded Spectra

9.3 Convergence of Distribution Functions

9.4 Continuous Distribution Functions: A General Integral Transform of a Characteristic Function

9.5 A Basic Theorem on Analytic Characteristic Functions

9.6 Continuity Theorems on Intervals and Uniqueness Theorems

9.7 The Compact Set of Characteristic Functions

Notes

X Some Properties of Characteristic Functions

10.1 Characteristic Properties of Fourier Coefficients

10.2 Basic Theorems on Characterization of a Characteristic Function

103. Characteristic Properties of Characteristic Functions

10.4 Functions of the Wiener Class

10.5 Some Sufficient Criteria for Characteristic Functions

10.6 More Criteria for Characteristic Functions

Notes

XI Distribution Functions and their Characteristic Functions

11.1 Moments, Basic Properties

11.2 Smoothness of a Characteristic Function and the Existence of Moments

11.3 More About Smoothness of Characteristic Functions and Existence of Moments

11.4 Absolute Moments

11.5 Boundedness of the Spectra of Distribution Functions

11.6 Integrable Characteristic Functions

11.7 Analyticity of Distribution Functions

11.8 Mean Concentration Function of a Distribution Function

11.9 Some Properties of Analytic Characteristic Functions

11.10 Characteristic Functions Analytic in the Half-Plane

11.11 Entire Characteristic Functions I

11.12 Entire Characteristic Functions II

Notes

XII Convergence of Series of Independent Random Variables

12.1 Convergence of a Sequence of Random Variables

12.2 The Borel Theorem

12.3 The Zero-One Law

12.4 The Equivalence Theorem

12.5 The Three Series Theorem

12.6 Sufficient Conditions for the Convergence of a Series

12.7 Convergence Criteria and the Typical Function

12.8 Rademacher and Steinhaus Functions

12.9 Convergence of Products of Characteristic Functions

12.10 Unconditional Convergence

12.11 Absolute Convergence

12.12 Essential Convergence

Notes

XIII Properties of Sums of Independent Random Variables; Convergence of Series in the Mean

13.1 Continuity and Discontinuity Properties of the Sum of a Series

13.2 Integrability of the Sum of a Series

13.3 Magnitude of the Characteristic Functions of the Sums of Series

13.4 Distribution Functions of the Sums of Rademacher Series; Characteristic Functions of Singular Distributions

13.5 Further Theorems on Rademacher Series

13.6 Sums of Independent Random Variables

13.7 Convergent Systems

13.8 Integrability of Sums of Series; Strong and Weak Convergences of Series

13.9 Vanishing of the Sum of a Series

13.10 Summability of Series

Notes

XIV Some Special Series of Random Variables

14.1 Fourier Series with Rademacher Coefficients

14.2 Random Fourier Series

14.3 Random Power Series, Convergence

14.4 Convergence of Random Power Series with Identically and Independently Distributed Random Coefficients

14.5 Analytic Continuation of Random Power Series

14.6 Fourier Series with Orthogonal Random Coefficients

Notes

References

Index

## Product details

- No. of pages: 680
- Language: English
- Copyright: © Academic Press 1972
- Published: January 28, 1972
- Imprint: Academic Press
- eBook ISBN: 9781483218526

## About the Author

### Tatsuo Kawata

## About the Editors

### Z. W. Birnbaum

### E. Lukacs

#### Affiliations and Expertise

Bowling Green State University

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