Geodesic
On minimal moment assumptions in Berry–Esséen theorems for $U$-statistics
Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 3, pp. 596-614 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 
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     author = {V. Bentkus and F. G\"otze},
     title = {On minimal moment assumptions in {Berry{\textendash}Ess\'een} theorems for $U$-statistics},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {596--614},
     year = {1995},
     volume = {40},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1995_40_3_a8/}
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V. Bentkus; F. Götze. On minimal moment assumptions in Berry–Esséen 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/