Bergström / Birnbaum / Lukacs Weak Convergence of Measures

Probability and Mathematical Statistics: A Series of Monographs and Textbooks
1. Auflage 2014
ISBN: 978-1-4831-9145-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 260 Seiten, Web PDF

Weak Convergence of Measures provides information pertinent to the fundamental aspects of weak convergence in probability theory. This book covers a variety of topics, including random variables, Hilbert spaces, Gaussian transforms, probability spaces, and random variables. Organized into six chapters, this book begins with an overview of elementary fundamental notions, including sets, different classes of sets, different topological spaces, and different classes of functions and measures. This text then provides the connection between functionals and measures by providing a detailed introduction of the abstract integral as a bounded, linear functional. Other chapters consider weak convergence of sequences of measures, such as convergence of sequences of bounded, linear functionals. This book discusses as well the weak convergence in the C- and D-spaces, which is reduced to limit problems. The final chapter deals with weak convergence in separable Hilbert spaces. This book is a valuable resource for mathematicians.

Autoren/Hrsg.

Bergström, Harald

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Weak Convergence of Measures;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;10
6;CHAPTER I. SPACES, MAPPINGS, AND MEASURES;14
6.1;1. CLASSES OF SETS;14
6.2;2. ALEXANDROV SPACES, TOPOLOGICAL SPACES, AND MEASURABLE SPACES;17
6.3;3. MAPPINGS;19
6.4;4. CLASSES OF BOUNDED, REAL-VALUED, CONTINUOUS FUNCTIONS AND MEASURABLE FUNCTIONS;24
6.5;5. NORMAL SPACES AND COMPLETELY NORMAL SPACES;28
6.6;6. SEQUENCES OF SETS;30
6.7;7. METRIC SPACES;33
6.8;8. MAPPINGS INTO METRIC SPACES;36
6.9;9. PRODUCT SPACES;38
6.10;10. PRODUCT SPACES OF INFINITELY MANY FACTORS;41
6.11;11. SOME PARTICULAR METRIC SPACES;43
6.12;12. MEASURES ON AN ALGEBRA OF SUBSETS;46
6.13;13. MEASURES ON A-SPACES;48
6.14;14. EXTENSIONS OF MEASURES;51
6.15;15. MEASURES ON INFINITE-DIMENSIONAL PRODUCT SPACES;54
6.16;16. COMPLETION OF MEASURES, CONTINUITY ALMOST SURELY AND ALMOST EVERYWHERE;57
7;CHAPTER II. INTEGRALS, BOUNDED, LINEAR FUNCTIONALS, AND MEASURES;60
7.1;1. INTEGRALS AS NONNEGATIVE, BOUNDED, LINEAR FUNCTIONALS;60
7.2;2. GENERALIZATIONS OF THE ABSTRACT INTEGRAL;63
7.3;3. THE REPRESENTATIONS OF BOUNDED, LINEAR FUNCTIONALS BY INTEGRALS;65
7.4;4. MEASURES BELONGING TO A NONNEGATIVE, BOUNDED, LINEAR FUNCTIONAL ON A NORMAL A-SPACE;67
7.5;5. TRANSFORMATIONS OF MEASURES AND INTEGRALS;70
7.6;6. CONSTRUCTIONS OF MEASURES ON METRIC SPACES BY RIEMANN-STIELTJES INTEGRALS;72
7.7;7. MEASURES ON PRODUCT SPACES;74
7.8;8. CONVOLUTIONS OF MEASURES;77
7.9;9. PROBABILITY SPACES AND RANDOM VARIABLES;79
7.10;10. EXPECTATIONS, CONDITIONAL EXPECTATIONS, AND CONDITIONAL PROBABILITIES;81
7.11;11. THE JENSEN INEQUALITY;86
8;CHAPTER III. WEAK CONVERGENCE IN NORMAL SPACES;90
8.1;1. WEAK CONVERGENCE OF SEQUENCES OF MEASURES ON NORMAL SPACES;90
8.2;2. WEAK CONVERGENCE OF SEQUENCES OF INDUCED MEASURES AND TRANSFORMED MEASURES;94
8.3;3. UNIFORMLY s-SMOOTH SEQUENCES OF MEASURES;95
8.4;4. WEAK LIMITS OF SEQUENCES OF s-SMOOTH MEASURES ON COMPLETELY NORMAL A-SPACES;98
8.5;5. REDUCTION OF WEAK LIMIT PROBLEMS BY TRANSFORMATIONS;101
8.6;6. THE REDUCTION PROCEDURE FOR METRIC SPACES;103
8.7;7. WEAK CONVERGENCE OF TIGHT SEQUENCES OF MEASURES ON METRIC SPACES;106
8.8;8. SEMINORMS ON AN ALGEBRA;108
8.9;9. SOME FUNDAMENTAL IDENTITIES AND INEQUALITIES FOR PRODUCTS;109
8.10;10. CONVERGENCE IN SEMINORMS OF POWERS TO INFINITELY DIVISIBLE ELEMENTS;112
8.11;11. CONVERGENCE IN SEMINORMS OF PRODUCTS;114
9;CHAPTER IV. WEAK CONVERGENCE ON R(k);116
9.1;1. s-SMOOTH MEASURES ON R(k);116
9.2;2. GAUSSIAN MEASURES AND GAUSSIAN TRANSFORMS;117
9.3;3. FOURIER TRANSFORMS AND THEIR RELATION TO GAUSSIAN TRANSFORMS;121
9.4;4. GAUSSIAN SEMINORMS;125
9.5;5. THE SEMIGROUP OF s-SMOOTH MEASURES;131
9.6;6. STABILITY CONDITIONS FOR CONVOLUTION PRODUCTS THAT CONVERGE WEAKLY;134
9.7;7. THE UNIQUE DIVISIBILITY OF INFINITELY DIVISIBLE s-SMOOTH MEASURES;138
9.8;8. LÉVY MEASURES ON R(k); GAUSSIAN FUNCTIONALS;139
9.9;9. WEAK CONVERGENCE OF CONVOLUTION POWERS OF s-SMOOTH MEASURES;143
9.10;10. THE SEMIGROUP OF INFINITELY DIVISIBLE s-SMOOTH MEASURES;147
9.11;11. THE CHARACTERISTIC FUNCTION OF AN INFINITELY DIVISIBLE PROBABILITY MEASURE ON R(k) AND ITS CONNECTION WITH THE GAUSSIAN FUNCTIONALS;152
9.12;12. WEAK CONVERGENCE OF CONVOLUTION PRODUCTS;155
9.13;13. STABLE PROBABILITY MEASURES;157
9.14;14. GAUSSIAN TRANSFORMS AND GAUSSIAN SEMINORMS OF RANDOM VARIABLES: A COMPARISON METHOD;164
9.15;15. WEAK LIMITS OF DISTRIBUTIONS OF SUMS OF MARTINGALE DIFFERENCES;168
9.16;16. WEAK LIMITS OF DISTRIBUTIONS OF SUMS OF RANDOM VARIABLES UNDER INDEPENDENCE AND f-MIXING;172
10;CHAPTER V. WEAK CONVERGENCE ON THE C- AND D-SPACES;180
10.1;1. THE C- AND D-SPACES;180
10.2;2. PROJECTIONS;183
10.3;3. APPROXIMATIONS OF FUNCTIONS BY SCHAUDER SEQUENCES;185
10.4;4. WEAK CONVERGENCE;190
10.5;5. FLUCTUATIONS AND WEAK CONVERGENCE;193
10.6;6. CONSTRUCTION OF PROBABILITY MEASURES ON THE C- AND D-SPACES;198
10.7;7. GAUSSIAN s-SMOOTH MEASURES ON THE C- AND D-SPACES;201
10.8;8. EMBEDDING OF SUMS OF REAL-VALUED RANDOM VARIABLES IN RANDOM FUNCTIONS INTO THE D-SPACE;204
10.9;9. EMPIRICAL DISTRIBUTION FUNCTIONS;210
10.10;10. EMBEDDING OF SEQUENCES OF MARTINGALE DIFFERENCES IN RANDOM FUNCTIONS;214
11;CHAPTER VI. WEAK CONVERGENCE IN SEPARABLE HILBERT SPACES;218
11.1;1. s-SMOOTH MEASURES ON l2-SPACE;218
11.2;2. WEAK CONVERGENCE OF CONVOLUTION PRODUCTS OF PROBABILITY MEASURES ON I;221
11.3;3. NECESSARY AND SUFFICIENT CONDITIONS FOR THE WEAK CONVERGENCE OF CONVOLUTION PRODUCTS OF SYMMETRICAL PROBABILITY MEASURES;225
11.4;4. NECESSARY AND SUFFICIENT CONDITIONS FOR THE WEAK CONVERGENCE OF CONVOLUTION POWERS OF PROBABILITY MEASURES;227
11.5;5. DIFFERENT FORMS OF NECESSARY AND SUFFICIENT CONDITIONS FOR THE WEAK CONVERGENCE OF CONVOLUTION POWERS OF PROBABILITY MEASURES ON I;232
11.6;6. INVARIANTS OF INFINITELY DIVISIBLE s-SMOOTH MEASURES ON l2. GAUSSIAN FUNCTIONALS;241
11.7;7. THE CHARACTERISTIC FUNCTION OF PROBABILITY MEASURES ON I;246
12;APPENDIX: A PRODUCT-SUM IDENTITY;248
13;NOTES AND COMMENTS;251
14;BIBLIOGRAPHY;254
15;INDEX;256