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On the Estimation of Multiple Random Integrals and U-Statistics

  • Book
  • © 2013

Overview

Authors:
  1. Péter Major

    1. Alfréd Rényi Mathematical Institute, Hungarian Academy of Sciences, Budapest, Hungary

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Part of the book series: Lecture Notes in Mathematics (LNM, volume 2079)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xiii
    Download chapter PDF

  2. Introduction

    • Péter Major
    Pages 1-3

  3. Motivation of the Investigation: Discussion of Some Problems

    • Péter Major
    Pages 5-13

  4. Some Estimates About Sums of Independent Random Variables

    • Péter Major
    Pages 15-20

  5. On the Supremum of a Nice Class of Partial Sums

    • Péter Major
    Pages 21-33

  6. Vapnik–Červonenkis Classes and L 2-Dense Classes of Functions

    • Péter Major
    Pages 35-39

  7. The Proof of Theorems 4.1 and 4.2 on the Supremum of Random Sums

    • Péter Major
    Pages 41-51

  8. The Completion of the Proof of Theorem 4.1

    • Péter Major
    Pages 53-64

  9. Formulation of the Main Results of This Work

    • Péter Major
    Pages 65-78

  10. Some Results About U-statistics

    • Péter Major
    Pages 79-95

  11. Multiple Wiener–Itô Integrals and Their Properties

    • Péter Major
    Pages 97-120

  12. The Diagram Formula for Products of Degenerate U-Statistics

    • Péter Major
    Pages 121-138

  13. The Proof of the Diagram Formula for U-Statistics

    • Péter Major
    Pages 139-149

  14. The Proof of Theorems 8.3, 8.5 and Example 8.7

    • Péter Major
    Pages 151-168

  15. Reduction of the Main Result in This Work

    • Péter Major
    Pages 169-179

  16. The Strategy of the Proof for the Main Result of This Work

    • Péter Major
    Pages 181-189

  17. A Symmetrization Argument

    • Péter Major
    Pages 191-208

  18. The Proof of the Main Result

    • Péter Major
    Pages 209-225

  19. An Overview of the Results and a Discussion of the Literature

    • Péter Major
    Pages 227-245

  20. Back Matter

    Pages 247-290
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Keywords

About this book

This work starts with the study of those limit theorems in probability theory for which classical methods do not work. In many cases some form of linearization can help to solve the problem, because the linearized version is simpler. But in order to apply such a method we have to show that the linearization causes a negligible error. The estimation of this error leads to some important large deviation type problems, and the main subject of this work is their investigation. We provide sharp estimates of the tail distribution of multiple integrals with respect to a normalized empirical measure and so-called degenerate U-statistics and also of the supremum of appropriate classes of such quantities. The proofs apply a number of useful techniques of modern probability that enable us to investigate the non-linear functionals of independent random variables.
This lecture note yields insights into these methods, and may also be useful for those who only want some new tools to help them prove limit theorems when standard methods are not a viable option.

Authors and Affiliations

  • Alfréd Rényi Mathematical Institute, Hungarian Academy of Sciences, Budapest, Hungary

    Péter Major

Bibliographic Information

  • Book Title: On the Estimation of Multiple Random Integrals and U-Statistics

  • Authors: Péter Major

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-642-37617-7

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2013

  • Softcover ISBN: 978-3-642-37616-0Published: 08 July 2013

  • eBook ISBN: 978-3-642-37617-7Published: 28 June 2013

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XIII, 288

  • Number of Illustrations: 11 b/w illustrations

  • Topics: Probability Theory and Stochastic Processes

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