Lower estimates of the convergence rate for \(U\)-statistics (Q1904476)

The aim of this paper is to show that the rate of convergence \(O(n^{-1/2})\) in the CLT for nondegenerate \(U\)-statistics (of degree 2) with symmetric kernel \(h(x,y)\) is not possible to obtain when the well-known moment conditions \(E |E [h(X_1, X_2) \mid X_1] |^3 < \infty\) and \(E |h(X_1, X_2) |^{5/3} < \infty\) are replaced by assuming the finiteness of moments of order \(< 3\) or \(< 5/3\), respectively. Note that some results on rates of convergence under weaker moment assumptions are given by the reviewer and \textit{W. Wolf} [J. Multivariate Anal. 44, No. 2, 266-278 (1993; Zbl 0768.60023)].