Parthasarathy / Birnbaum / Lukacs Probability Measures on Metric Spaces

1. Auflage 2014
ISBN: 978-1-4832-2525-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 288 Seiten, Web PDF

Probability Measures on Metric Spaces presents the general theory of probability measures in abstract metric spaces. This book deals with complete separable metric groups, locally impact abelian groups, Hilbert spaces, and the spaces of continuous functions. Organized into seven chapters, this book begins with an overview of isomorphism theorem, which states that two Borel subsets of complete separable metric spaces are isomorphic if and only if they have the same cardinality. This text then deals with properties such as tightness, regularity, and perfectness of measures defined on metric spaces. Other chapters consider the arithmetic of probability distributions in topological groups. This book discusses as well the proofs of the classical extension theorems and existence of conditional and regular conditional probabilities in standard Borel spaces. The final chapter deals with the compactness criteria for sets of probability measures and their applications to testing statistical hypotheses. This book is a valuable resource for statisticians.

Autoren/Hrsg.

Parthasarathy, K. R.

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Fron Cover;1
2;Probability Measures on Metric Spaces;4
3;Copyright Page;5
4;Table of Contents;10
5;Preface;6
6;Chapter
I. The Borel Subsets of a Metric Space;14
6.1;1. General Properties of Borel Sets;14
6.2;2. The Isomorphism Theorem;20
6.3;3. The Kuratowski Theorem;28
6.4;4. Borel Cross Sections in Compact Metric Spaces;35
6.5;5. Borel Cross Sections in Locally Compact Groups;37
7;Chapter
II. Probability Measures in a Metric Space;39
7.1;1. Regular Measures;39
7.2;2. Spectrum of a Measure;40
7.3;3. Tight Measures;41
7.4;4. Perfect Measures;43
7.5;5. Linear Functionals and Measures;45
7.6;6. The Weak Topology in the Space of Measures;52
7.7;7. Convergence of Sample Distributions;65
7.8;8. Existence of Nonatomic Measures in Metric Spaces;66
8;Chapter
III. Probability Measures in a Metric Group;69
8.1;1. The Convolution Operation;69
8.2;2. Shift Compactness in M(X);71
8.3;3. Idempotent Measures;74
8.4;4. Indecomposable Measures;76
8.5;5. The Case When X Is Abelian;83
9;Chapter VI.
Probability Measures in Locally Compact Abelian Groups;86
9.1;1. Introduction;86
9.2;2. Preliminary Facts about a Group and Its Character Group ;87
9.3;3. Measures and Their Fourier Transforms;87
9.4;4. Infinitely Divisible Distributions;90
9.5;5. General Limit Theorems for Sums of Infinitesimal Summands ;95
9.6;6. Gaussian Distributions;110
9.7;7. Representation of Infinitely Divisible Distributions;115
9.8;8. Uniqueness of the Representation;122
9.9;9. Compactness Criteria;126
9.10;10. Representation of Convolution Semigroups;129
9.11;11. A Decomposition Theorem;131
9.12;12. Absolutely Continuous Indecomposable Distributions in X ;133
10;Chapter
V. The Kolmogorov Consistency Theorem and Conditional Probability;144
10.1;1. Statement of the First Problem;144
10.2;2. Standard Borel Spaces;145
10.3;3. The Consistency Theorem in the Case of Inverse Limits of Borel Spaces;150
10.4;4. The Extension Theorem;153
10.5;5. The Kolmogorov Consistency Theorem;156
10.6;6. Statement of the Second Problem;157
10.7;7. Existence of Conditional Probability;158
10.8;8. Regular Conditional Probability;159
11;Chapter
VI. Probability Measures in a Hilbert Space;164
11.1;1. Introduction;164
11.2;2. Characteristic Functions and Compactness Criteria;164
11.3;3. An Estimate of the Variance;178
11.4;4. Infinitely Divisible Distributions;183
11.5;5. Compactness Criteria;195
11.6;6. Accompanying Laws;202
11.7;7. Representation of Convolution Semigroups;214
11.8;8. Decomposition Theorem;214
11.9;9. Ergodic Theorems;215
12;Chapter
VII. Probability Measures on C[0, 1] and D[0, 1];224
12.1;1. Introduction;224
12.2;2. Probability Measures on C[0, 1];225
12.3;3. A Condition for the Realization of a Stochastic Process in
C;228
12.4;4. Convergence to Brownian Motion;232
12.5;5. Distributions of Certain Random Variables Associated with the Brownian Motion;237
12.6;7. Probability Measures in
D;262
12.7;8. Ergodic Theorems for D-Valued Random Variables;267
12.8;9. Applications to Statistical Tests of Hypothesis;272
13;Bibliographical Notes;281
14;Bibliography;283
15;List of Symbols;286
16;Author Index;287
17;Subject Index;288