Geodesic
Lyapunov-Type Bounds for $U$-Statistics
Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 4, pp. 724-743

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Let $X_1,\dots,X_n$ be independent identically distributed random variables. An optimal Lyapunov (or Berry–Esseen) bound is derived for $U$-statistics of degree 2, that is, statistics of the form $\sum_{j$, where $H$ is a measurable, symmetric function such that $\mathbf{E}\,|H(X_1,X_2)|\infty$, assuming that the statistic is nondegenerate.
Keywords: $U$-statistics, Lyapunov-type bound, Berry–Esseen bound, rate of convergence, normal approximations. 
@article{TVP_2001_46_4_a6,
     author = {I. B. Alberink and V. Yu. Bentkus},
     title = {Lyapunov-Type {Bounds} for $U${-Statistics}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {724--743},
     publisher = {mathdoc},
     volume = {46},
     number = {4},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2001_46_4_a6/}
}
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I. B. Alberink; V. Yu. Bentkus. Lyapunov-Type Bounds for $U$-Statistics. Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 4, pp. 724-743. http://geodesic.mathdoc.fr/item/TVP_2001_46_4_a6/