On a multivariate version of Bernstein’s inequality. (English) Zbl 1127.60017

Major, Péter

Electron. J. Probab. 12, 966-988 (2007).

Summary: We prove 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 is compared with an analogous bound about the tail distribution of multiple Wiener-Itô 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.

MSC:

60E15	Inequalities; stochastic orderings
60F10	Large deviations
60H99	Stochastic analysis

Keywords:

degenerate \(U\)-statistics; Wiener-Itô integrals; high moments; diagram formula