On minimal moment assumptions in Berry–Esséen theorems for $U$-statistics

The rate of convergence for asymptotically normal $U$-statistics is of order $O(n^{-1/2})$ provided that
$$ \mathbf{E}|\mathbf{E}\{h(X_1,X_2)\mid X_1\}|^3<\infty \quad\text{and}\quad \mathbf{E}|h(X_1,X_2)|^{5/3}<\infty, $$
where $h$ is a symmetric kernel corresponding to the $U$-statistic. Bentkus, Götze, and Zitikis [preprint 92-075, Universität Bielefeld, 1992] have shown that the last moment condition is the best possible, that is, it cannot be replaced by a moment of order $\frac53-\varepsilon$, for any $\varepsilon>0$. In this paper we extend the result for statistics of higher orders and with possible nonnormal limit distributions.