Tail behaviour of multiple random integrals and U-statistics

Abstract

This paper contains sharp estimates about the distribution of multiple random integrals of functions of several variables with respect to a normalized empirical measure, about the distribution of U-statistics and multiple Wiener-Ito integrals with respect to a white noise. It also contains good estimates about the supremum of appropriate classes of such integrals or U-statistics. The proof of most results is omitted, I have concentrated on the explanation of their content and the picture behind them. I also tried to explain the reason for the investigation of such questions. My goal was to yield such a presentation of the results which a non-expert also can understand, and not only on a formal level.

Published at http://dx.doi.org/10.1214/154957805100000186 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Estimates of moments and tails of Gaussian chaoses
• Moment inequalities for U-statistics
• A multivariate generalization of Hoeffding's inequality
• An estimate on the maximum of a nice class of stochastic integrals
• On a multivariate version of Bernstein's inequality

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• Asymptotic Expansion of Gaussian Chaos via Probabilistic Approach
• Empirical limit theorems for Wiener chaos
• Exchangeable Hoeffding decompositions over finite sets: a characterization and counterexamples