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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 forpolynomials of independent Gaussian random variables. The proof is based on theso-called diagram formula for Wiener--It\^o integrals which yields a goodrepresentation for their products as a sum of such integrals. I intend to showin a subsequent paper that this method also yields good estimates fordegenerate $U$-statistics. The main result of this paper is a generalization ofthe estimates of Hanson and Wright about bilinear forms of independent standardnormal random variables. On the other hand, it is a weaker estimate than themain result of a paper of Lata{\l}a [6]. But that paper contains an error, andit is not clear whether its result is true. This question is also discussedhere.

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