Balasubrahmanyan / Lau Functional Equations in Probability Theory

1. Auflage 2014
ISBN: 978-1-4832-7222-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 268 Seiten, Web PDF

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.

Autoren/Hrsg.

Balasubrahmanyan, Ramachandran

Lau, Ka-Sing

Inhaltsverzeichnis

1;Front Cover;1
2;Functional Equations in Probability Theory;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;Preface;12
7;Introduction;14
8;Chapter 1. Background Material;20
8.1;1.1. Cauchy Functional Equations;20
8.2;1.2. Auxiliary Results from Analysis;25
8.3;1.3. Distribution Functions and Characteristic Functions;29
8.4;Notes and Remarks;41
9;Chapter 2. Integrated Cauchy Functional Equations on IR+;42
9.1;2.1. The ICFE on Z+;43
9.2;2.2. The ICFE on IR+;45
9.3;2.3. An Alternative Proof Using Exchangable R.V.'S;52
9.4;2.4. The ICFE with a Signed Measure;54
9.5;2.5. Application to Characterization of Probability Distributions;58
9.6;Notes and Remarks;68
10;Chapter 3. The Stable Laws, the Semistable Laws, and a Generalization;70
10.1;3.1. The Stable Laws;70
10.2;3.2. The Semistable Laws;78
10.3;3.3. The Generalized Semistable Laws and the Normal Solutions;80
10.4;3.4. The Generalized Semistable Laws and the Nonnormal Solutions;82
10.5;Appendix: Series Expansions for Stable Densities (a . 1, 2);88
10.6;Notes and Remarks;89
11;Chapter 4. Integrated Cauchy Functional Equations with Error Terms on IR+;90
11.1;4.1. ICFE's with Error Terms on IR+: The First Kind;90
11.2;4.2. Characterizations of Weibull Distribution;94
11.3;4.3. A Characterization of Semistable Laws;100
11.4;4.4. ICFE's with Error Terms on IR+: The Second Kind;109
11.5;Notes and Remarks;116
12;Chapter 5. Independent/Identically Distributed Linear Forms, and the Normal Laws;117
12.1;5.1. Identically Distributed Linear Forms;118
12.2;5.2. Proof of the Sufficiency Part of Linnik's Theorem;124
12.3;5.3. Proof of the Necessity Part of Linnik's Theorem;132
12.4;5.4. Zinger's Theorem;138
12.5;5.5. Independence of Linear Forms in Independent R.V.'S;144
12.6;Notes and Remarks;147
13;Chapter 6. Independent/Identical Distribution Problems Relating to Stochastic Integrals;149
13.1;6.1. Stochastic Integrals;149
13.2;6.2. Characterization of Wiener Processes;153
13.3;6.3. Identically Distributed Stochastic Integrals and Stable Processes;157
13.4;6.4. Identically Distributed Stochastic Integrals and Semistable Processes;168
13.5;Appendix: Some Phragmén-Lindelöf-type Theorems and Other Auxiliary Results;173
13.6;Notes and Remarks;177
14;Chapter 7. Distribution Problems Relating to the Arc-sine, the Normal, and the Chi-Square Laws;178
14.1;7.1. An Equidistribution Problem, and the Arc-sine Law;178
14.2;7.2. Distribution Problems Involving the Normal and the x12 Laws;188
14.3;7.3. Quadratic Forms, Noncentral x2 Laws, and Normality;193
14.4;Notes and Remarks;203
15;Chapter 8. Integrated Cauchy Functional Equations on IR;204
15.1;8.1. The ICFE on IR and on Z;204
15.2;8.2. A Proof Using the Krein-Milman Theorem;216
15.3;8.3. A Variant of the ICFE on IR and the Wiener-Hopf Technique;218
15.4;Notes and Remarks;228
16;Chapter 9. Integrated Cauchy Functional Equations on Semigroups of IRd;229
16.1;9.1. Exponential Functions on Semigroups;230
16.2;9.2. Translations of Measures;235
16.3;9.3. The Skew Convolution;239
16.4;9.4. The Cones Defined by Convolutions and Their Extreme Rays;243
16.5;9.5. The ICFE on Semigroups of IRd;249
16.6;Appendix: Weak Convergence of Measures; Choquet's Theorem;255
16.7;Notes and Remarks;258
17;Bibliography;260
18;Index;266
19;PROBABILITY AND MATHEMATICAL STATISTICS;270