Probability measures on metric spaces / K. R. Parthasarathy.

Format:

Book

Author/Creator:

Parthasarathy, K. R., author.

Series:

AMS Chelsea Publishing Series

AMS Chelsea Publishing, v. 352

Language:

English

Subjects (All):

Probability measures.

Metric spaces.

Physical Description:

1 online resource (290 pages)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : AMS Chelsea Publishing, 2014.

Summary:

In this book, the author gives a cohesive account of the theory of probability measures on complete metric spaces (which is viewed as an alternative approach to the general theory of stochastic processes). After a general description of the basics of topology on the set of measures, the author discusses regularity, tightness, and perfectness of measures, properties of sampling distributions, and metrizability and compactness theorems. Next, he describes arithmetic properties of probability measures on metric groups and locally compact abelian groups. Covered in detail are notions such as decomposability, infinite divisibility, idempotence, and their relevance to limit theorems for "sums" of infinitesimal random variables. The book concludes with numerous results related to limit theorems for probability measures on Hilbert spaces and on the space of continuous functions on an interval.This book is suitable for graduate students and researchers interested in probability and stochastic processes and would make an ideal supplementary reading or independent study text.

Contents:

Front Cover

PREFACE

CONTENTS

I THE BOREL SUBSETS OF A METRIC SPACE

1. GENERAL PROPERTIES OF BOREL SETS

2. THE ISOMORPHISM THEOREM

3. THE KURATOWSKI THEOREM

4. BOREL CROSS SECTIONS IN COMPACT METRIC SPACES

5. BOREL CROSS SECTIONS IN LOCALLY COMPACT GROUPS

II PROBABILITY MEASURES IN A METRIC SPACE

I. REGULAR MEASURES

2. SPECTRUM OF A MEASURE

3. TIGHT MEASURES

4. PERFECT MEASURES

5. LINEAR FUNCTIONALS AND MEASURES

6. THE WEAK TOPOLOGY IN THE SPACE OF MEASURES

7. CONVERGENCE OF SAMPLE DISTRIBUTIONS

8. EXISTENCE OF NONATOMIC MEASURES IN METRIC SPACES

III PROBABILITY MEASURES IN A METRIC GROUP

1. THE CONVOLUTION OPERATION

2. SHIFT COMPACTNESS IN M(X)

3. IDEMPOTENT MEASURES

4. INDECOMPOSABLE MEASURES

5. THE CASE WHEN X IS ABELIAN

IV PROBABILITY MEASURES IN LOCALLY COMPACT ABELIAN GROUPS

1. INTRODUCTION

2. PRELIMINARY FACTS ABOUT A GROUP AND ITS CHARACTER GROUP

3. MEASURES AND THEIR FOURIER TRANSFORMS

4. INFINITELY DIVISIBLE DISTRIBUTIONS

5. GENERAL LIMIT THEOREMS FOR SUMS OF INFINITESIMAL SUMMANDS

6. GAUSSIAN DISTRIBUTIONS

7. REPRESENTATION OF INFINITELY DIVISIBLE DISTRIBUTIONS

8. UNIQUENESS OF THE REPRESENTATION

9. COMPACTNESS CRITERIA

10. REPRESENTATION OF CONVOLUTION SEMI GROUPS

11. A DECOMPOSITION THEOREM

12. ABSOLUTELY CONTINUOUS INDECOMPOSABLE DISTRIBUTIONS IN X

V THE KOLMOGOROV CONSISTENCY THEOREM AND CONDITIONAL PROBABILITY

1. STATEMENT OF THE FIRST PROBLEM

2. STANDARD BOREL SPACES

3. THE CONSISTENCY THEOREM IN THE CASE OF INVERSE LIMITS OF BOREL SPACES

4. THE EXTENSION THEOREM

5. THE KOLMOGOROV CONSISTENCY THEOREM

6. STATEMENT OF THE SECOND PROBLEM

7. EXISTENCE OF CONDITIONAL PROBABILITY

8. REGULAR CONDITIONAL PROBABILITY

VI PROBABILITY MEASURES IN A HILBERT SPACE

1. INTRODUCTION.

2. CHARACTERISTIC FUNCTIONS AND COMPACTNESS CRITERIA

3. AN ESTIMATE OF THE VARIANCE

5. COMPACTNESS CRITERIA

6. ACCOMPANYING LAWS

7. REPRESENTATION OF CONVOLUTION SEMIGROUPS

8. DECOMPOSITION THEOREM

9. ERGODIC THEOREMS

VII PROBABILITY MEASURES ON C[0,1] AND D[0,1]

I. INTRODUCTION

2. PROBABILITY MEASURES ON c[0,1]

3. A CONDITION FOR THE REALIZATION OF A STOCHASTIC PROCESS IN C

4. CONVERGENCE TO BROWNIAN MOTION

5. DISTRIBUTIONS OF CERTAIN RANDOM VARIABLES ASSOCIATED WITH THE BROWNIAN MOTION

6. THE SPACE D[0,1]

7. PROBABILITY MEASURES IN D

8. ERGODIC THEOREMS FORD-VALUED RANDOM VARIABLES

9. APPLICATIONS TO STATISTICAL TESTS OF HYPOTHESIS

BIBLIOGRAPHICAL NOTES

BIBLIOGRAPHY

LIST OF SYMBOLS

AUTHOR INDEX

SUBJECT INDEX

Back Cover.

Notes:

Originally published: New York : Academic Press, 1967, in series: Probability and mathematical statistics, a series of monographs and textbooks.

Includes bibliographical references and indexes.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

1-4704-3028-2

OCLC:

982018861