On the estimation of multiple random integrals and U-statistics / Péter Major

Auteur: Major, Péter (1947-) - AuteurType de document: MonographieCollection: Lecture notes in mathematics ; 2079Langue: anglaisPays: AllemagneÉditeur: Berlin : Springer, cop. 2013Description: 1 vol. (XIII-288 p.) ; 24 cm ISBN: 9783642376160 ; br. Résumé: 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. (Source : Springer).Bibliographie: Bibliogr. p. 283-285. Index. Sujets MSC: 60-02 Probability theory and stochastic processes -- Research exposition (monographs, survey articles)
60H05 Probability theory and stochastic processes -- Stochastic analysis -- Stochastic integrals
62G20 Statistics -- Nonparametric inference -- Asymptotic properties
60F10 Probability theory and stochastic processes -- Limit theorems -- Large deviations
60G60 Probability theory and stochastic processes -- Stochastic processes -- Random fields
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Bibliogr. p. 283-285. Index

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. (Source : Springer)

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