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Uniform Central Limit Theorems

2nd Edition

$124.00 (P)

Part of Cambridge Studies in Advanced Mathematics

  • Date Published: February 2014
  • availability: Available
  • format: Hardback
  • isbn: 9780521498845

$ 124.00 (P)
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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.

    • 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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    Product details

    • Edition: 2nd Edition
    • Date Published: February 2014
    • format: Hardback
    • isbn: 9780521498845
    • length: 486 pages
    • dimensions: 229 x 152 x 32 mm
    • weight: 0.77kg
    • availability: Available
  • Table of Contents

    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.

  • Author

    R. M. Dudley, Massachusetts Institute of Technology
    R. M. Dudley is a Professor of Mathematics at the Massachusetts Institute of Technology in Cambridge, Massachusetts.

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