Synopses & Reviews
This book shows how, when samples become large, the probability laws of large numbers and related facts are guaranteed to hold over wide domains. 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 Giné-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 recent 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 with an interest in probability, mathematical statisticians, and computer scientists working in computer learning theory.
Review
"This monograph is a well-written treatise on functional central limit theorems...recommended for all who want to know more about a subject which by now is considered a must in abstract large-sample theory." Mathematical Reviews
Synopsis
This treatise by an acknowledged expert includes several topics not found in any previous book.
Synopsis
The central limit theorem shows how, when samples become large, the probability laws of large numbers and related facts are guaranteed to hold over wide domains. This thorough treatise on the subject, by an acknowledged expert, includes several topics not found in any other book, such as the treatment of VC combinatorics, the proofs of a bootstrap central limit theorem and of invariance principles. It also includes problems at the end of each chapter. The book will interest mathematicians working in probability, mathematical statisticians, and computer scientists working in computer learning theory.
Table of Contents
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-Červonenkis combinatorics; 5. Measurability; 6. Limit theorems for Vapnik-Červonenkis 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.