Functional Equations in Probability Theory - 1st Edition - ISBN: 9780124377301, 9781483272221

Functional Equations in Probability Theory

1st Edition

Authors: Ramachandran Balasubrahmanyan Ka-Sing Lau
eBook ISBN: 9781483272221
Imprint: Academic Press
Published Date: 28th September 1991
Page Count: 268
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Functional Equations in Probability Theory deals with functional equations in probability theory and covers topics ranging from the integrated Cauchy functional equation (ICFE) to stable and semistable laws. The problem of identical distribution of two linear forms in independent and identically distributed random variables is also considered, with particular reference to the context of the common distribution of these random variables being normal.

Comprised of nine chapters, this volume begins with an introduction to Cauchy functional equations as well as distribution functions and characteristic functions. The discussion then turns to the nonnegative solutions of ICFE on R+; ICFE with a signed measure; and application of ICFE to the characterization of probability distributions. Subsequent chapters focus on stable and semistable laws; ICFE with error terms on R+; independent/identically distributed linear forms and the normal laws; and distribution problems relating to the arc-sine, the normal, and the chi-square laws. The final chapter is devoted to ICFE on semigroups of Rd.

This book should be of interest to mathematicians and statisticians.

Table of Contents



Chapter 1 Background Material

1.1. Cauchy Functional Equations

1.2. Auxiliary Results from Analysis

1.3. Distribution Functions and Characteristic Functions

Notes and Remarks

Chapter 2 Integrated Cauchy Functional Equations on R+

2.1. The ICFE on Z+

2.2. The ICFE on R+

2.3. An Alternative Proof Using Exchangable R.V.'s

2.4. The ICFE with a Signed Measure

2.5. Application to Characterization of Probability Distributions

Notes and Remarks

Chapter 3 The Stable Laws, the Semistable Laws, and a Generalization

3.1. The Stable Laws

3.2. The Semistable Laws

3.3. The Generalized Semistable Laws and the Normal Solutions

3.4. The Generalized Semistable Laws and the Nonnormal Solutions

Appendix: Series Expansions for Stable Densities (α ≠ 1,2)

Notes and Remarks

Chapter 4 Integrated Cauchy Functional Equations with Error Terms on R+

4.1. ICFE's with Error Terms on R+.: The First Kind

4.2. Characterizations of Weibull Distribution

4.3. A Characterization of Semistable Laws

4.4. ICFE's with Error Terms on R+.: The Second Kind

Notes and Remarks

Chapter 5 Independent/Identically Distributed Linear Forms, and the Normal Laws

5.1. Identically Distributed Linear Forms

5.2. Proof of the Sufficiency Part of Linnik's Theorem

5.3. Proof of the Necessity Part of Linnik's Theorem

5.4. Zinger's Theorem

5.5. Independence of Linear Forms in Independent R.V.'s

Notes and Remarks

Chapter 6 Independent/Identical Distribution Problems Relating to Stochastic Integrals

6.1. Stochastic Integrals

6.2. Characterization of Wiener Processes

6.3. Identically Distributed Stochastic Integrals and Stable Processes

6.4. Identically Distributed Stochastic Integrals and Semistable Processes

Appendix: Some Phragmen-Lindelöf-type Theorems and Other Auxiliary Results

Notes and Remarks

Chapter 7 Distribution Problems Relating to the Arc-sine, the Normal, and the Chi-Square Laws

7.1. An Equidistribution Problem, and the Arc-sine Law

7.2. Distribution Problems Involving the Normal and the χ21 Laws

7.3. Quadratic Forms, Noncentral χ2 Laws, and Normality

Notes and Remarks

Chapter 8 Integrated Cauchy Functional Equations on R

8.1. The ICFE on R and on Z

8.2. A Proof Using the Krein-Milman Theorem

8.3. A Variant of the ICFE on R and the Wiener-Hopf Technique

Notes and Remarks

Chapter 9 Integrated Cauchy Functional Equations on Semigroups of Rd

9.1. Exponential Functions on Semigroups

9.2. Translations of Measures

9.3. The Skew Convolution

9.4. The Cones Defined by Convolutions and Their Extreme Rays

9.5. The ICFE on Semigroups of Rd

Appendix: Weak Convergence of Measures; Choquet's Theorem

Notes and Remarks




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© Academic Press 1991
Academic Press
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About the Author

Ramachandran Balasubrahmanyan

Ka-Sing Lau

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