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 Probability Surveys > Vol. 2 (2005) open journal systems 


Tail behaviour of multiple random integrals and $U$-statistics

Péter Major, Alfred Renyi Mathematical Institute of the Hungarian Academy of Sciences


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–Itô 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.

AMS 2000 subject classifications: Primary 60F10; secondary 60G50.

Keywords: multiple Wiener--Itô integrals, (degenerate) $U$-statistics, large-deviation type estimates, diagram formula, symmetrization, decoupling method, $L_p$-dense classes of functions.

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Major, Péter, Tail behaviour of multiple random integrals and $U$-statistics, Probability Surveys, 2, (2005), 448-505 (electronic).

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Probability Surveys. ISSN: 1549-5787