2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73212We prove a multivariate version of Bernstein's inequality about the probability that degenerate $U$-statistics take a value larger than some number $u$. This is an improvement of former estimates for the same problem which yields an asymptotically sharp estimate for not too large numbers $u$. This paper also contains an analogous bound about the distribution of multiple Wiener-Ito integrals. Their comparison shows that our results are sharp. The proofs are based on good estimates about high moments of multiple random integrals. They are obtained by means of a diagram formula which enables us to express the product of multiple random integrals as the sum of such expressions.Probability60E15text