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

TL;DR

This paper provides estimates for moments and tail distributions of Wiener--Itô integrals and Gaussian polynomial functions, generalizing Hanson-Wright bounds and discussing related errors in prior work.

Contribution

It introduces a new method using diagram formulas to estimate Wiener--Itô integrals and Gaussian polynomials, extending Hanson-Wright type inequalities.

Findings

Derived bounds for moments and tail probabilities of Wiener--Itô integrals

Extended Hanson-Wright estimates to polynomials of Gaussian variables

Discussed errors in previous related results

Abstract

In this paper I prove good estimates on the moments and tail distribution of $k$-fold Wiener--It\^o 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--It\^o 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 Lata{\l}a [6]. But that paper contains an error, and it is not clear whether its result is true. This question is also discussed here.