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Berry–Esseen bounds for von Mises and U-statistics

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Abstract

Let X 1,... ,X n be i.i.d.\ random variables. An optimal Berry–Esseen bound is derived for U-statistics of order 2, that is, statistics of the form ∑ j> k H (X j , X k ), where H is a measurable, symmetric function such that E | H (X1, X2)| > ∞, assuming that the statistic is non-degenerate. The same is done for von Mises statistics, that is, statistics of the form ∑ j,k H (X j , X k ). As a corollary, the central limit theorem is derived under optimal moment conditions.

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Authors and Affiliations

  1. Department of Mathematics, University of Nijmegen, Postbus 9010, 6500 GL, Nijmegen, The Netherlands

    I. B. Alberink & V. Bentkus

  2. Institute of Mathematics and Informatics, Akademijos 4, 2600, Vilnius, Lithuania

    V. Bentkus

Authors
  1. I. B. Alberink
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  2. V. Bentkus
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Alberink, I.B., Bentkus, V. Berry–Esseen bounds for von Mises and U-statistics. Lithuanian Mathematical Journal 41, 1–16 (2001). https://doi.org/10.1023/A:1011066719481

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