On the Estimation of Multiple Random Integrals and U-StatisticsThis 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. |
Contents
| 2 | |
| 5 | |
| 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 |
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On the Estimation of Multiple Random Integrals and U-Statistics Péter Major No preview available - 2013 |
Common terms and phrases
ˇ ˇ ˇ Ä j Ä applied Banach space central limit theorem Chap class of functions coloured diagrams constant decoupled U-statistics defined in formula degenerate U-statistics denotes diagram formula distributed random variables exponent f 2 F function f function f(x1 Gaussian random field Hence Hermite polynomials Hilbert space Hoeffding decomposition holds identically distributed random implies In;k f independent random variables integrals with respect introduced kernel function L2-dense class L2-norm Lecture Notes Lemma Let us consider measurable functions measurable space X;X multiple random integrals multiple Wiener–Itˆo integrals multivariate version non-atomic non-atomic measure normalized empirical distribution normalized empirical measure parameter polynomials probability measure problem proof of Proposition proof of Theorem Proposition 15.3 Proposition 6.1 proved random field real numbers relation result right-hand side sequence of independent subsets sup f2F supremum tail behaviour tail distribution term Theorem 4.1 Vapnik–ˇCervonenkis classes vectors white noise Wiener—Itó integrals