Characterization of Probability Distributions on Locally Compact Abelian Groups

Contents Preface Chapter I. Preliminaries 1. Results of abstract harmonic analysis 2. Probability distributions on topological Abelian groups 3. Gaussian distributions on locally compact Abelian groups Chapter II. Independent random variables with independent sum and difference 4. Identically distributed random variables 5. General case Chapter III. Characterization of probability distributions through the independence of linear forms 6. General characterization theorems 7. Shifts of idempotent distributions on discrete and compact totally disconnected Abelian groups 8. Gaussian distributions on the cylinder R×T Chapter IV. Characterization of probability distributions through the symmetry of the conditional distribution of one linear form given another 9. Locally compact Abelian groups containing no elements of order 2 10. Discrete and compact totally disconnected Abelian groups 11. Locally compact Abelian groups containing an element of order 2 Chapter V. Characterization theorems on the field of p-adic numbers 12. Skitovich–Darmois theorem 13. Heyde theorem 14. Characterization of shifts of idempotent distributions through the independence of sum and difference squared Chapter VI. Miscellaneous characterization theorems 15. Rao theorems 16. Generalized Pólya theorem Bibliography Index of terms Index of symbols Contents 6 Preface 8 Chapter I. Preliminaries 12 1. Results of abstract harmonic analysis 12 2. Probability distributions on topological Abelian groups 21 3. Gaussian distributions on locally compact Abelian groups 26 Chapter II. Independent random variables with independent sum and difference 38 4. Identically distributed random variables 38 5. General case 59 Chapter III. Characterization of probability distributions through the independence of linear forms 74 6. General characterization theorems 74 7. Shifts of idempotent distributions on discrete and compact totally disconnected Abelian groups 90 8. Gaussian distributions on the cylinder R×T 108 Chapter IV. Characterization of probability distributions through the symmetry of the conditional distribution of one linear form given another 124 9. Locally compact Abelian groups containing no elements of order 2 124 10. Discrete and compact totally disconnected Abelian groups 145 11. Locally compact Abelian groups containing an element of order 2 166 Chapter V. Characterization theorems on the field of p-adic numbers 196 12. Skitovich–Darmois theorem 196 13. Heyde theorem 201 14. Characterization of shifts of idempotent distributions through the independence of sum and difference squared 211 Chapter VI. Miscellaneous characterization theorems 224 15. Rao theorems 224 16. Generalized Pólya theorem 233 Bibliography 242 Index of terms 248 Index of symbols 250