Estimation of Wiener--Ito integrals and polynomials of independent Gaussian random variables

Peter Major

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arXiv · 0803.1453

Estimation of Wiener--Ito integrals and polynomials of independent Gaussian random variables

Peter Major

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Abstract

In this paper I prove good estimates on the moments and tail distribution of $k$-fold Wiener--Itô integrals and also present their natural counterpart for polynomials of independent Gaussian random variables. The proof is based on the so-called diagram formula for Wiener--Itô integrals which yields a good representation for their products as a sum of such integrals. I intend to show in a subsequent paper that this method also yields good estimates for degenerate $U$-statistics. The main result of this paper is a generalization of the estimates of Hanson and Wright about bilinear forms of independent standard normal random variables. On the other hand, it is a weaker estimate than the main result of a paper of Latała [6]. But that paper contains an error, and it is not clear whether its result is true. This question is also discussed here.

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Peter Major. 2008-03-10. Estimation of Wiener--Ito integrals and polynomials of independent Gaussian random variables. https://arxiv.org/abs/0803.1453

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