Lower Estimates of the Convergence Rate for $U$-Statistics

Abstract

Recent results on the Berry-Esseen bound for $U$-statistics assumed the following conditions: Suppose a $U$-statistic (of degree 2) is nondegenerate. Then the rate of convergence in the CLT is of the order $O(n^{-1/2})$ provided that $\mathbb{E}|\mathbb{E}\{h(X_1, X_2)|X_1\}|^3 < \infty, \mathbb{E}|h(X_1, X_2)|^{5/3} < \infty,$ where $h$ is a symmetric kernel corresponding to the $U$-statistic. It follows from our results that these moment conditions are final. In particular, the last moment condition cannot be replaced by a moment of order $5/3 - \epsilon$ for any $\epsilon > 0$. Similar results hold for von Mises statistics.