Approximations for multivariate U-statistics (Q578787)

Let \(X_ 1,X_ 2,..\). be independent, identically distributed random variables with values in a measurable space (\({\mathcal X},{\mathcal F})\) and let \(H: {\mathcal X}\times {\mathcal X}\to {\mathbb{R}}^ k\) be measurable and symmetric. The author derives a Berry-Esseen type approximation and an Edgeworth expansion of order up to \(o(N^{-})\) of the distribution of \(2^{- 1}N^{1/2}U_ N\), where \(U_ N\) denotes the k-variate U-statistic of degree 2 defined by H and the \(X_ k's\) improving and extending earlier results by various authors. The main assumption here is E \(\| H(X_ 1,X_ 2)\|^ 2<\infty\).