Uniform Central Limit Theorems

This expanded edition of the classic work on empirical processes now boasts several new proved theorems not in the first.

R. M. Dudley (Author)

9780521498845, Cambridge University Press

Hardback, published 24 February 2014

486 pages
23.5 x 15.8 x 3 cm, 0.77 kg

In this new edition of a classic work on empirical processes the author, an acknowledged expert, gives a thorough treatment of the subject with the addition of 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, 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.

1. Donsker's theorem and inequalities
2. Gaussian processes, sample continuity
3. Definition of Donsker classes
4. Vapnik–Cervonenkis combinatorics
5. Measurability
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.

Subject Areas: Stochastics [PBWL], Combinatorics & graph theory [PBV], Probability & statistics [PBT], Calculus & mathematical analysis [PBK]