On minimal moment assumptions in Berry--Ess\'een theorems for $U$-statistics
Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 3, pp. 596-614
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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.
Keywords:
U-statistics Berry–Esseén bound, lower bound.
Mots-clés : convergence rate
Mots-clés : convergence rate
@article{TVP_1995_40_3_a8,
author = {V. Bentkus and F. G\"otze},
title = {On minimal moment assumptions in {Berry--Ess\'een} theorems for $U$-statistics},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {596--614},
publisher = {mathdoc},
volume = {40},
number = {3},
year = {1995},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1995_40_3_a8/}
}
TY - JOUR AU - V. Bentkus AU - F. Götze TI - On minimal moment assumptions in Berry--Ess\'een theorems for $U$-statistics JO - Teoriâ veroâtnostej i ee primeneniâ PY - 1995 SP - 596 EP - 614 VL - 40 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TVP_1995_40_3_a8/ LA - ru ID - TVP_1995_40_3_a8 ER -
V. Bentkus; F. Götze. On minimal moment assumptions in Berry--Ess\'een theorems for $U$-statistics. Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 3, pp. 596-614. http://geodesic.mathdoc.fr/item/TVP_1995_40_3_a8/