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.
Author(s): Harald Bergström
Series: Probability and Mathematical Statistics
Publisher: Academic Press
Year: 1982
Language: English
Pages: xii+245
Preface
Chapter I Spaces, Mappings, and Measures
1. Classes of Sets
2. Alexandrov Spaces, Topological Spaces, and Measurable Spaces
3. Mappings
4. Classes of Bounded, Real-Valued, Continuous Functions and Measurable Functions
5. Normal Spaces and Completely Normal Spaces
6. Sequences of Sets
7. Metric Spaces
8. Mappings into Metric Spaces
9. Product Spaces
10. Product Spaces of Infinitely Many Factors
11. Some Particular Metric Spaces
12. Measures on an Algebra of Subsets
13. Measures on A-Spaces
14. Extensions of Measures
15. Measures on Infinite-Dimensional Product Spaces
16. Completion of Measures, Continuity Almost Surely and Almost Everywhere
Chapter II Integrals, Bounded, Linear Functionals, and Measures
1. Integrals as Nonnegative, Bounded, Linear Functionals
2. Generalizations of the Abstract Integral
3. The Representations of Bounded, Linear Functionals by Integrals
4. Measures Belonging to a Nonnegative, Bounded, Linear Functional on a Normal A-Space
5. Transformations of Measures and Integrals
6. Constructions of Measures on Metric Spaces by Riemann-Stieltjes Integrals
7. Measures on Product Spaces
8. Convolutions of Measures
9. Probability Spaces and Random Variables
10. Expectations, Conditional Expectations, and Conditional Probabilities
11. The Jensen Inequality
Chapter III Weak Convergence in Normal Spaces
1. Weak Convergence of Sequences of Measures on Normal Spaces
2. Weak Convergence of Sequences of Induced Measures and Transformed Measures
3. Uniformly s-Smooth Sequences of Measures
4. Weak Limits of s-Smooth Measures on Completely Normal A-Spaces
5. Reduction of Weak Limit Problems by Transformations
6. The Reduction Procedure for Metric Spaces
7. Weak Convergence of Tight Sequences of Measures on Metric Spaces
8. Seminorms on an Algebra
9. Some Fundamental Identities and Inequalities for Products
10. Convergence in Seminorms of Powers to Infinitely Divisible Elements
11. Convergence in Seminorms of Products
Chapter IV Weak Convergence ON R(k)
1. s-Smooth Measures on R(k)
2. Gaussian Measures and Gaussian Transforms
3. Fourier Transforms and Their Relation to Gaussian Transforms
4. Gaussian Seminorms
5. The Semigroup of s-Smooth Measures
6. Stability Conditions for Convolution Products That Converge Weakly
7. The Unique Divisibility of Infinitely Divisible s-Smooth Measures
8. Lévy Measures on R(k); Gaussian Functionals
9. Weak Convergence of Convolution Powers of s-Smooth Measures
10. The Semigroup of Infinitely Divisible s-Smooth Measures
11. The Characteristic Function of an Infinitely Divisible Probability Measure on R{k) and Its Connection with the Gaussian Functional
12. Weak Convergence of Convolution Products
13. Stable Probability Measures
14. Gaussian Transforms and Gaussian Seminorms of Random Variables: A Comparison Method
15. Weak Limits of Distributions of Sums of Martingale Differences
16. Weak Limits of Distributions of Sums of Random Variables under Independence and f-Mixing
Chapter V Weak Convergence on the C- and D-Spaces
1. The C- and D-Spaces
2. Projections
3. Approximations of Functions by Schauder Sequences
4. Weak Convergence
5. Fluctuations and Weak Convergence
6. Construction of Probability Measures on the C- and D-Spaces
7. Gaussian s-Smooth Measures on the C- and D-Spaces
8. Embedding of Sums of Real-Valued Random Variables in Random Functions into the D-Space
9. Empirical Distribution Functions
10. Embedding of Sequences of Martingale Differences in Random Functions
Chapter VI Weak Convergence in Separable Hilbert Spaces
1. s-Smooth Measures on l2-Space
2. Weak Convergence of Convolution Products of Probability Measures on l2
3. Necessary and Sufficient Conditions for the Weak Convergence of Convolution Products of Symmetrical Probability Measures
4. Necessary and Sufficient Conditions for the Weak Convergence of Convolution Powers of Probability Measures
5. Different Forms of Necessary and Sufficient Conditions for the Weak Convergence of Convolution Powers of Probability Measures on l2
6. Invariants of Infinitely Divisible s-Smooth Measures on l2 Gaussian Functionals
7. The Characteristic Function of Probability Measures on l2
Appendix
A Product-Sum Identity
Notes and Comments
Bibliography
Index
