On the Estimation of Multiple Random Integrals and U-Statistics

1. Front Matter

   Pages i-xiii

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2. Introduction

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   Pages 1-3

3. Motivation of the Investigation: Discussion of Some Problems

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   Pages 5-13

4. Some Estimates About Sums of Independent Random Variables

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   Pages 15-20

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

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   Pages 21-33

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

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   Pages 35-39

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

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   Pages 41-51

8. The Completion of the Proof of Theorem 4.1

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   Pages 53-64

9. Formulation of the Main Results of This Work

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   Pages 65-78

10. Some Results About U-statistics

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    Pages 79-95

11. Multiple Wiener–Itô Integrals and Their Properties

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    Pages 97-120

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

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    Pages 121-138

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

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    Pages 139-149

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

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    Pages 151-168

15. Reduction of the Main Result in This Work

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    Pages 169-179

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

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    Pages 181-189

17. A Symmetrization Argument

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    Pages 191-208

18. The Proof of the Main Result

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    Pages 209-225

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

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    Pages 227-245

20. Back Matter

    Pages 247-290

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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.

• 60H05, 60G60, 60F10
• Chaining argument
• Large deviation estimates
• Vapnik–Červonenkis classes
• Wiener–Itô integrals and diagram formula

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

  Péter Major

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