CiNii 図書 - Uniform central limit theorems

This book shows how the central limit theorem for independent, identically distributed random variables with values in general, multidimensional spaces, holds uniformly over some large classes of functions. The author, an acknowledged expert, gives a thorough treatment of the subject, including several topics not found in any previous book, such as the Fernique-Talagrand majorizing measure theorem for Gaussian processes, an extended treatment of Vapnik-Chervonenkis combinatorics, the Ossiander L2 bracketing central limit theorem, the Gine-Zinn bootstrap central limit theorem in probability, the Bronstein theorem on approximation of convex sets, and the Shor theorem on rates of convergence over lower layers. Other results of Talagrand and others are surveyed without proofs in separate sections. Problems are included at the end of each chapter so the book can be used as an advanced text. The book will interest mathematicians working in probability, mathematical statisticians and computer scientists working in computer learning theory.

Preface
1. Introduction: Donsker's theorem, metric entropy and inequalities
2. Gaussian measures and processes
sample continuity
3. Foundations of uniform central limit theorems: Donsker classes
4. Vapnik-Cervonenkis combinatorics
5. Measurability
6. Limit theorems for Vapnik-Cervonenkis and related classes
7. Metric entropy, with inclusion and bracketing
8. Approximation of functions and sets
9. Sums in general Banach spaces and invariance principles
10. Universal and uniform central limit theorems
11. The two-sample case, the bootstrap, and confidence sets
12. Classes of sets or functions too large for central limit theorems
Appendices
Subject index
Author index
Index of notation.

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Uniform central limit theorems

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