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 Electronic Journal of Probability > Vol. 12 (2007) > Paper 34 open journal systems 


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.


Full text: PDF

Pages: 966-988

Published on: August 2, 2007
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Electronic Journal of Probability. ISSN: 1083-6489