On a Multivariate Version of Bernstein's Inequality

Abstract

We prove such a multivariate version of Bernstein's inequality about the tail distribution of degenerate $U$-statistics which is an improvement of some former results. This estimate will be compared with an analogous bound about the tail distribution of multiple Wiener-Ito integrals. Their comparison shows that our estimate is sharp. The proof is based on good estimates about high moments of degenerate $U$-statistics. They are obtained by means of a diagram formula which enables us to express the product of degenerate $U$-statistics as the sum of such expressions.