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Teor. Veroyatnost. i Primenen., 2001, Volume 46, Issue 4, Pages 724–743 (Mi tvp3797)  

This article is cited in 5 scientific papers (total in 5 papers)

Lyapunov-Type Bounds for $U$-Statistics

I. B. Alberinka, V. Yu. Bentkusb

a University of Nijmegen, Department of Mathematics
b Institute of Mathematics and Informatics

Abstract: Let $X_1,…,X_n$ 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

DOI: https://doi.org/10.4213/tvp3797

Full text: PDF file (1609 kB)

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

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Received: 26.01.2000
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Citation: I. B. Alberink, V. Yu. Bentkus, “Lyapunov-Type Bounds for $U$-Statistics”, Teor. Veroyatnost. i Primenen., 46:4 (2001), 724–743; Theory Probab. Appl., 46:4 (2002), 571–588

Citation in format AMSBIB
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\paper Lyapunov-Type Bounds for $U$-Statistics
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\jour Theory Probab. Appl.
\yr 2002
\vol 46
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\pages 571--588
\crossref{https://doi.org/10.1137/S0040585X97979299}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Jing Bing-Yi, Wang Qiying, “Edgeworth expansion for $U$-statistics under minimal conditions”, Ann. Statist., 31:4 (2003), 1376–1391  crossref  mathscinet  zmath  isi  scopus
    2. Wang Qiying, Weber N.C., “Exact convergence rate and leading term in the central limit theorem for $U$-statistics”, Statist. Sinica, 16:4 (2006), 1409–1422  mathscinet  zmath  isi  elib
    3. Bentkus V., Jing Bing-Yi, Zhou Wang, “On normal approximations to $U$-statistics”, Ann. Probab., 37:6 (2009), 2174–2199  crossref  mathscinet  zmath  isi  scopus
    4. Lai T.L., Shao Q.-M., Wang Q., “Cramer Type Moderate Deviations for Studentized U-Statistics”, ESAIM-Probability and Statistics, 15 (2011), 168–179  crossref  mathscinet  zmath  isi  scopus
    5. Shao Q.-M., Zhou W.-X., “Cram?r type moderate deviation theorems for self-normalized processes”, Bernoulli, 22:4 (2016), 2029–2079  crossref  mathscinet  zmath  isi  elib  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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