Probability Measures on Metric Spaces (eBook)

K. R. Parthasarathy (Autor)

Z. W. Birnbaum, E. Lukacs (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
288 Seiten
Elsevier Science (Verlag)
978-1-4832-2525-8 (ISBN)

Infos

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.

Inhalt

Fron Cover 1
Probability Measures on Metric Spaces 4
Copyright Page 5
Table of Contents 10
Preface 6
Chapter 
14 
1. General Properties of Borel Sets 14
2. The Isomorphism Theorem 20
3. The Kuratowski Theorem 28
4. Borel Cross Sections in Compact Metric Spaces 35
5. Borel Cross Sections in Locally Compact Groups 37
Chapter 
39 
1. Regular Measures 39
2. Spectrum of a Measure 40
3. Tight Measures 41
4. Perfect Measures 43
5. Linear Functionals and Measures 45
6. The Weak Topology in the Space of Measures 52
7. Convergence of Sample Distributions 65
8. Existence of Nonatomic Measures in Metric Spaces 66
Chapter 
69 
1. The Convolution Operation 69
2. Shift Compactness in M(X) 71
3. Idempotent Measures 74
4. Indecomposable Measures 76
5. The Case When X Is Abelian 83
Chapter VI. 
86 
1. Introduction 86
2. Preliminary Facts about a Group and Its Character Group 87
3. Measures and Their Fourier Transforms 87
4. Infinitely Divisible Distributions 90
5. General Limit Theorems for Sums of Infinitesimal Summands 95
6. Gaussian Distributions 110
7. Representation of Infinitely Divisible Distributions 115
8. Uniqueness of the Representation 122
9. Compactness Criteria 126
10. Representation of Convolution Semigroups 129
11. A Decomposition Theorem 131
12. Absolutely Continuous Indecomposable Distributions in X 133
Chapter 
144 
1. Statement of the First Problem 144
2. Standard Borel Spaces 145
3. The Consistency Theorem in the Case of Inverse Limits of Borel Spaces 150
4. The Extension Theorem 153
5. The Kolmogorov Consistency Theorem 156
6. Statement of the Second Problem 157
7. Existence of Conditional Probability 158
8. Regular Conditional Probability 159
Chapter 
164 
1. Introduction 164
2. Characteristic Functions and Compactness Criteria 164
3. An Estimate of the Variance 178
4. Infinitely Divisible Distributions 183
5. Compactness Criteria 195
6. Accompanying Laws 202
7. Representation of Convolution Semigroups 214
8. Decomposition Theorem 214
9. Ergodic Theorems 215
Chapter 
224 
1. Introduction 224
2. Probability Measures on C[0, 1] 225
3. A Condition for the Realization of a Stochastic Process in 
228 
4. Convergence to Brownian Motion 232
5. Distributions of Certain Random Variables Associated with the Brownian Motion 237
7. Probability Measures in 
262 
8. Ergodic Theorems for D-Valued Random Variables 267
9. Applications to Statistical Tests of Hypothesis 272
Bibliographical Notes 281
Bibliography 283
List of Symbols 286
Author Index 287
Subject Index 288

Produkt-Details

• Erscheint lt. Verlag: 3.7.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• ISBN-10: 1-4832-2525-9 / 1483225259
• ISBN-13: 978-1-4832-2525-8 / 9781483225258