Open Access
2007 On a Multivariate Version of Bernstein's Inequality
Peter Major
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Electron. J. Probab. 12: 966-988 (2007). DOI: 10.1214/EJP.v12-430


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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Peter Major. "On a Multivariate Version of Bernstein's Inequality." Electron. J. Probab. 12 966 - 988, 2007.


Accepted: 2 August 2007; Published: 2007
First available in Project Euclid: 1 June 2016

zbMATH: 1127.60017
MathSciNet: MR2336595
Digital Object Identifier: 10.1214/EJP.v12-430

Primary: 60E15 , 60F10
Secondary: 60H99

Keywords: (degenerate) U-statistics , Bernstein inequality , diagram formula , moment estimates , Wiener--It^o integrals

Vol.12 • 2007
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