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This classic work on empirical processes has been considerably expanded and revised from the original edition. When samples become large, the probability laws of large numbers and central limit theorems are guaranteed to hold uniformly over wide domains. The author, an acknowledged expert, gives a thorough treatment of the subject, including 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. This new edition contains several proved theorems not included in the first edition, including the Bretagnolle-Massart theorem giving constants in the Komlos-Major-Tusnady rate of convergence for the classical empirical process, Massart's form of the Dvoretzky-Kiefer-Wolfowitz inequality with precise constant, Talagrand's generic chaining approach to boundedness of Gaussian processes, a characterization of uniform Glivenko-Cantelli classes of functions, Giné and Zinn's characterization of uniform Donsker classes (i.e., classing Donsker uniformly over all probability measures P), and the Bousquet-Koltchinskii-Panchenko theorem that the convex hull of a uniform Donsker class is uniform Donsker. The book will be an essential reference for mathematicians working in infinite-dimensional central limit theorems, mathematical statisticians, and computer scientists working in computer learning theory. Problems are included at the end of each chapter so the book can also be used as an advanced text.Read more
- A thoroughly revised second edition that includes updates and expansion of every chapter
- Includes a number of new proofs that were missing from the first edition
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- Edition: 2nd Edition
- Date Published: February 2014
- format: Paperback
- isbn: 9780521738415
- length: 482 pages
- dimensions: 228 x 152 x 26 mm
- weight: 0.65kg
- availability: In stock
Table of Contents
1. Donsker's theorem and inequalities
2. Gaussian processes, sample continuity
3. Definition of Donsker classes
4. Vapnik–Cervonenkis combinatorics
6. Limit theorems for VC-type classes
7. Metric entropy with bracketing
8. Approximation of functions and sets
9. Two samples and the bootstrap
10. Uniform and universal limit theorems
11. Classes too large to be Donsker
Appendix A. Differentiating under an integral sign
Appendix B. Multinomial distributions
Appendix C. Measures on nonseparable metric spaces
Appendix D. An extension of Lusin's theorem
Appendix E. Bochner and Pettis integrals
Appendix F. Non-existence of some linear forms
Appendix G. Separation of analytic sets
Appendix H. Young–Orlicz spaces
Appendix I. Versions of isonormal processes.
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