Operator-limit distributions in probability theory

✍ Scribed by Jurek Z.J., Mason J.D.

Publisher: Wiley
Year: 1993
Tongue: English
Leaves: 308
Series: Wiley Series in Probability and Statistics
Category: Library

No coin nor oath required. For personal study only.

✦ Synopsis

Written by two experts of multidimensional developments in a classic area of probability theory—the central limit theory. Features all essential tools to bring readers up to date in the field. Describes operator-selfdecomposable measures, operator-stable distributions and provides specialized techniques from probability theory.

✦ Table of Contents

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
Symbols and Notation......Page 8
Preface......Page 10
1.1 Numakura Theorem,......Page 16
1.2 Linear Spaces,......Page 18
1.3 Integration and Differentiation of Vector-Valued Functions,......Page 22
1.4 One-Parameter Semigroups,......Page 25
1.5 Basic Facts from Lie Theory,......Page 30
1.6 Probability Measures on Metric Spaces,......Page 35
1.7 Probabilities on Banach Spaces,......Page 38
1.8 Infinitely Divisible Probabilities,......Page 43
1.9 The Skorohod Space,......Page 54
1.10 Bibliographic Comments,......Page 59
2. Convergence of Types Theorems, Symmetry Groups, and Decomposability Semigroups,......Page 60
2.1 Full Measures,......Page 61
2.2 Convergence of Types Theorems,......Page 63
2.3 Urbanik Decomposability Semigroup,......Page 68
2.4 Compact Subgroups of cal(A)_l(R^d) and Aut(R^d),......Page 74
2.5 Examples in Infinite-Dimensional Spaces,......Page 77
2.6 The Case When d = 1; Comments,......Page 80
2.7 Standard Convergence of Types,......Page 82
2.8 Bibliographic Comments,......Page 85
3.1 Statement of the Problem,......Page 86
3.2 Norming Sequences,......Page 87
3.3 The Urbanik Semigroup of an Operator- Selfdecomposable Measure and Its Exponents,......Page 91
3.4 Characteristic Functionals of exp(- tQ)- Decomposable Measures,......Page 104
3.5 Generators of the Class L_0(Q),......Page 123
3.6 Random Integral Representation,......Page 131
3.7 Infinitesimal Generators,......Page 159
3.8 The Absolute Continuity of exp(- tQ)- Decomposable Measures,......Page 177
3.9 Multivariate Selfdecomposable Measures,......Page 192
3.10 Bibliographic Comments,......Page 197
4.1 Statement of the Problem,......Page 200
4.2 Sakovic-Sharpe Characterization,......Page 201
4.3 A Class of tB-Stable Measures,......Page 208
4.4 The Class of .(B) and Fix Points of the Mapping TB,......Page 213
4.5 Norming Sequences,......Page 216
4.6 Structural Characterizations of Operator-Stable Measures and Their Exponents,......Page 220
4.7 Commuting Exponents and Other Norms,......Page 234
4.8 Elliptically Symmetric Operator-Stable Measures,......Page 244
4.9 The Centering Function NO,......Page 251
4.10 Absolute Continuity of Full Operator-Stable Measures,......Page 252
4.11 Domains of Normal Attraction of Operator-Stable Measures,......Page 254
4.12 Moments,......Page 264
4.13 Independent Marginals,......Page 269
4.14 Multivariate Stable Measures,......Page 276
4.15 The Case When d = 3,......Page 286
4.16 Bibliographic Comments,......Page 289
Epilogue......Page 292
Bibliography......Page 296
Index......Page 304
Back Cover......Page 308

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