On the Estimation of Multiple Random Integrals and U-Statistics

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Springer, Jun 28, 2013 - Mathematics - 288 pages
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.
 

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Contents

Chapter 1 Introduction
2
Discussion of Some Problems
5
Chapter 3 Some Estimates About Sums of Independent Random Variables
15
Chapter 4 On the Supremum of a Nice Class of Partial Sums
21
Chapter 5 VapnikČervonenkis Classes and L2Dense Classes of Functions
34
Chapter 6 The Proof of Theorems 41 and 42 on the Supremum of Random Sums
41
Chapter 7 The Completion of the Proof of Theorem 41
52
Chapter 8 Formulation of the Main Results of This Work
65
Chapter 14 Reduction of the Main Result in This Work
169
Chapter 15 The Strategy of the Proof for the Main Result of This Work
181
Chapter 16 A Symmetrization Argument
190
Chapter 17 The Proof of the Main Result
209
Chapter 18 An Overview of the Results and a Discussion of the Literature
227
Appendix A The Proof of Some Results About VapnikČervonenkis Classes
246
Appendix B The Proof of the Diagram Formula for WienerItô Integrals
251
Appendix C The Proof of Some Results About WienerItô Integrals
261

Chapter 9 Some Results About Ustatistics
79
Chapter 10 Multiple WienerItô Integrals and Their Properties
97
Chapter 11 The Diagram Formula for Products of Degenerate UStatistics
121
Chapter 12 The Proof of the Diagram Formula for UStatistics
139
Chapter 13 The Proof of Theorems 83 85 and Example 87
150
Appendix D The Proof of Theorem 143 About UStatistics and Decoupled UStatistics
270
References
283
Index
287
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