Functional Equations in Probability Theory (eBook)

Ramachandran Balasubrahmanyan, Ka-Sing Lau (Autoren)

eBook Download: PDF

2014 | 1. Auflage
268 Seiten
Elsevier Science (Verlag)
978-1-4832-7222-1 (ISBN)

Infos

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.

Inhalt

Front Cover 1
Functional Equations in Probability Theory 4
Copyright Page 5
Table of Contents 8
Dedication 6
Preface 12
Introduction 14
Chapter 1. Background Material 20
1.1. Cauchy Functional Equations 20
1.2. Auxiliary Results from Analysis 25
1.3. Distribution Functions and Characteristic Functions 29
Notes and Remarks 41
Chapter 2. Integrated Cauchy Functional Equations on IR+ 42
2.1. The ICFE on Z+ 43
2.2. The ICFE on IR+ 45
2.3. An Alternative Proof Using Exchangable R.V.'S 52
2.4. The ICFE with a Signed Measure 54
2.5. Application to Characterization of Probability Distributions 58
Notes and Remarks 68
Chapter 3. The Stable Laws, the Semistable Laws, and a Generalization 70
3.1. The Stable Laws 70
3.2. The Semistable Laws 78
3.3. The Generalized Semistable Laws and the Normal Solutions 80
3.4. The Generalized Semistable Laws and the Nonnormal Solutions 82
Appendix: Series Expansions for Stable Densities (a . 1, 2) 88
Notes and Remarks 89
Chapter 4. Integrated Cauchy Functional Equations with Error Terms on IR+ 90
4.1. ICFE's with Error Terms on IR+: The First Kind 90
4.2. Characterizations of Weibull Distribution 94
4.3. A Characterization of Semistable Laws 100
4.4. ICFE's with Error Terms on IR+: The Second Kind 109
Notes and Remarks 116
Chapter 5. Independent/Identically Distributed Linear Forms, and the Normal Laws 117
5.1. Identically Distributed Linear Forms 118
5.2. Proof of the Sufficiency Part of Linnik's Theorem 124
5.3. Proof of the Necessity Part of Linnik's Theorem 132
5.4. Zinger's Theorem 138
5.5. Independence of Linear Forms in Independent R.V.'S 144
Notes and Remarks 147
Chapter 6. Independent/Identical Distribution Problems Relating to Stochastic Integrals 149
6.1. Stochastic Integrals 149
6.2. Characterization of Wiener Processes 153
6.3. Identically Distributed Stochastic Integrals and Stable Processes 157
6.4. Identically Distributed Stochastic Integrals and Semistable Processes 168
Appendix: Some Phragmén-Lindelöf-type Theorems and Other Auxiliary Results 173
Notes and Remarks 177
Chapter 7. Distribution Problems Relating to the Arc-sine, the Normal, and the Chi-Square Laws 178
7.1. An Equidistribution Problem, and the Arc-sine Law 178
7.2. Distribution Problems Involving the Normal and the x12 Laws 188
7.3. Quadratic Forms, Noncentral x2 Laws, and Normality 193
Notes and Remarks 203
Chapter 8. Integrated Cauchy Functional Equations on IR 204
8.1. The ICFE on IR and on Z 204
8.2. A Proof Using the Krein-Milman Theorem 216
8.3. A Variant of the ICFE on IR and the Wiener-Hopf Technique 218
Notes and Remarks 228
Chapter 9. Integrated Cauchy Functional Equations on Semigroups of IRd 229
9.1. Exponential Functions on Semigroups 230
9.2. Translations of Measures 235
9.3. The Skew Convolution 239
9.4. The Cones Defined by Convolutions and Their Extreme Rays 243
9.5. The ICFE on Semigroups of IRd 249
Appendix: Weak Convergence of Measures Choquet's Theorem
Notes and Remarks 258
Bibliography 260
Index 266
PROBABILITY AND MATHEMATICAL STATISTICS 270

Produkt-Details

• Erscheint lt. Verlag: 12.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• ISBN-10: 1-4832-7222-2 / 1483272222
• ISBN-13: 978-1-4832-7222-1 / 9781483272221