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.
Highlights
Let us consider a sequence of iid. random variables ξ1, ξ2, . . ., on a measurable space (X, X ) with some distribution μ together with a real valued function f = f (x1, . . . , xk) of k variables defined on the k-th power (Xk, X k) of the space (X, X ) and define with their help the U -statistics In,k(f ), n = k, k + 1, . . ., In,k(f ) = 1 k!f . (1)1≤js≤n, s=1,...,k js=js′ if s=s′We want to get good estimates on the probabilities P n−k/2k!|In,k(f )| > u for u > 0 under appropriate conditions.Let me first recall a result of Arcones and Gine (2) in this direction
Our goal is to prove such an improvement of this result which gives the right value of the parameter c2 in formula (2), and we want to explain the probabilistic content of such an improvement
2 2 we may restrict our attention to this case.) But while the variance and expectation determines the distribution of a Gaussian random variable, the distribution of a Wiener–Itointegral is not determined by its variance and expectation
Summary
If the canonical function f of k variables satisfies condition (4), the degenerate U -statistics n−k/2In,k(f ) converge in distribution to the k-fold Wiener–Ito integral Jμ,k(f ), Jμ,k(f ). The above result suggests to describe the tail-distribution of the Wiener–Ito integral Jμ,k(f ) and to show that Theorem 1 gives such an estimate which the above mentioned limit theorem and the tail distribution of Jμ,k(f ) suggests At this moment there appears an essential difference between the problem discussed in Bernstein’s inequality and its multivariate version. (Let me remark that the integral in formula (4) equals the variance of (k!)1/2Jμ,k(f ), and it is asymptotically equal to the variance of (k!)1/2n−k/2In,k(f ) for large n At least, this is the case if f is a symmetric function of its variables.
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