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-It^o 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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