Functional Equations in Probability Theory

PrefaceIntroductionChapter 1 Background Material 1.1. Cauchy Functional Equations 1.2. Auxiliary Results from Analysis 1.3. Distribution Functions and Characteristic Functions Notes and RemarksChapter 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 RemarksChapter 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 RemarksChapter 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 RemarksChapter 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 RemarksChapter 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 RemarksChapter 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 RemarksChapter 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 RemarksChapter 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 RemarksBibliographyIndex