Lower Estimates of the Convergence Rate for $U$-Statistics
V. Bentkus, Friedrich Götze, Ričardas Zitikis · The Annals of Probability · 1994
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 $\mathbb{E}|\mathbb{E}\{h(X_1, X_2)|X_1\}|^3 0$. Similar results hold for von Mises statistics.