This article is cited in 6 papers

Lyapunov-Type Bounds for $U$-Statistics

I. B. Alberink^a, V. Yu. Bentkus<sup>b

a</sup> University of Nijmegen, Department of Mathematics
^b Institute of Mathematics and Informatics

Abstract: 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 $\sum_{j<k}H(X_j,X_k)$, 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.

Received: 26.01.2000

Language: English

DOI: 10.4213/tvp3797

 English version: Theory of Probability and its Applications, 2002, 46:4, 571–588

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