Berry–Esseen bounds for von Mises and U-statistics

Abstract

Let X1,... ,Xn be i.i.d.\ random variables. An optimal Berry–Esseen bound is derived for U-statistics of order 2, that is, statistics of the form ∑j> kH (Xj, Xk), where H is a measurable, symmetric function such that E | H (X1, X2)| > ∞, assuming that the statistic is non-degenerate. The same is done for von Mises statistics, that is, statistics of the form ∑j,kH (Xj, Xk). As a corollary, the central limit theorem is derived under optimal moment conditions.