On a multivariate version of Bernstein's inequality

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We prove a multivariate version of Bernstein's inequality about the probability that degenerate U-statistics take a value larger than some number u. This is an improvement of former estimates for the same problem which yields an asymptotically sharp estimate for not too large numbers u. This paper also contains an analogous bound about the distribution of multiple Wiener-Ito integrals. Their comparison shows that our results are sharp. The proofs are based on good estimates about high moments of multiple random integrals. They are obtained by means of a diagram formula which enables us to express the product of multiple random integrals as the sum of such expressions.