On a Multivariate Version of Bernstein's Inequality

Peter Major (Renyi Mathematical Institute of the Hungarian Academy of Sciences)

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.

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Pages: 966-988

Publication Date: August 2, 2007

DOI: 10.1214/EJP.v12-430

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