Uniform Central Limit Theorems

by R. M. Dudley

• ISBN13: 9780521052214
• ISBN10: 0521052211
• Edition: Revised
• Format: Paperback
• Copyright: 2008-02-04
• Publisher: Cambridge University Press

Summary

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 Ginè-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.

Table of Contents

Preface

Introduction: Donskers theorem, metric entropy and inequalities

Gaussian measures and processes, sample continuity

Foundations of uniform central limit theorems: Donsker classes

Vapnik-ervonenkis combinatorics

Measurability

Limit theorems for Vapnik-ervonenkis and related classes

Metric entropy, with inclusion and bracketing

Approximation of functions and sets

Sums in general Banach spaces and invariance principles

Universal and uniform central limit theorems

The two-sample case, the bootstrap, and confidence sets

Classes of sets or functions too large for central limit theorems

Appendices

Subject index

Author index

Index of notation

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