Berry-Esseen bounds for von Mises and \(U\)-statistics (Q1347380)

An optimal Berry-Esseen type bound is derived for non-degenerate U-statistics of degree 2 and for related von Mises statistics. The bounds derived apply to the broader family of statistics where the kernel and underlying distributions depend on the sample size. For this broader class of statistics bounds are given for the normal approximation and the normal approximation with a one term Edgeworth correction. Further, these results have as a corollary optimal moment conditions for the central limit theorem for these statistics.