PUB - Publikationen an der Universität Bielefeld

The rate of convergence for asymptotically normal U-statistics is of order O(n(-1/2)) provided that E\E{h(X(1), X(2))/X(1)}\(3) < co and E\h(X(1), X(2))\(5/3) < co, where h is a symmetric kernel corresponding to the U-statistic. Bentkus, Gotze, and Zitikis [preprint 92-075, Universitat Bielefeld, 1992] have shown that the last moment condition is the best possible, that is, it cannot be replaced by a moment of order 5/3 - epsilon, for any epsilon > 0. In this paper we extend the result for statistics of higher orders and with possible nonnormal limit distributions.