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Sib. Èlektron. Mat. Izv., 2020, Volume 17, Pages 672–682 (Mi semr1240)  

This article is cited in 2 scientific papers (total in 2 papers)

Probability theory and mathematical statistics

Normal approximation for $U$- and $V$-statistics of a stationary absolutely regular sequence

V. G. Mikhailova, N. M. Mezhennayab

a Steklov Mathematical Institute of Russian Academy of Sciences, 8, Gubkina str., Moscow, 119991, Russia
b Bauman Moscow State Technical University, 5/1, 2-aya Baumanskaya str., Moscow, 105005, Russia

Abstract: Let $(X_{n,t})_{t=1}^{\infty}$ be a stationary absolutely regular sequence of real random variables with the distribution dependent on the number $n$. The paper presents sufficient conditions for the asymptotic normality (for $n\to\infty$ and common centering and normalization) of the distribution of the nonhomogeneous $U$-statistic of order $r$ which is given on the sequence $X_{n,1},\ldots,X_{n,n}$ with a kernel also dependent on $n$. The same results for $V$-statistics also hold. To analyze sums of dependent random variables with rare strong dependencies, the proof uses the approach that was proposed by S. Janson in 1988 and upgraded by V. Mikhailov in 1991 and M. Tikhomirova and V. Chistyakov in 2015.

Keywords: absolute regularity condition, characterizing graph, central limit theorem, dependency graph, $U$-statistic, $V$-statistic, stationary sequence.

DOI: https://doi.org/10.33048/semi.2020.17.045

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Bibliographic databases:

UDC: 519.214
MSC: 60F05, 05C90, 94C15
Received October 16, 2019, published May 8, 2020
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Citation: V. G. Mikhailov, N. M. Mezhennaya, “Normal approximation for $U$- and $V$-statistics of a stationary absolutely regular sequence”, Sib. Èlektron. Mat. Izv., 17 (2020), 672–682

Citation in format AMSBIB
\Bibitem{MikMez20}
\by V.~G.~Mikhailov, N.~M.~Mezhennaya
\paper Normal approximation for $U$- and $V$-statistics of a stationary absolutely regular sequence
\jour Sib. \`Elektron. Mat. Izv.
\yr 2020
\vol 17
\pages 672--682
\mathnet{http://mi.mathnet.ru/semr1240}
\crossref{https://doi.org/10.33048/semi.2020.17.045}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000532336600001}


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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. V. G. Mikhailov, N. M. Mezhennaya, A. V. Volgin, “Ob usloviyakh asimptoticheskoi normalnosti chisla povtorenii v statsionarnoi sluchainoi posledovatelnosti”, Diskret. matem., 33:3 (2021), 64–78  mathnet  crossref
    2. N. M. Mezhennaya, V. G. Mikhailov, “Tsentralnaya predelnaya teorema dlya $U$-statistik ot tsepochek metok vershin na polnom grafe”, PDM. Prilozhenie, 2021, no. 14, 30–32  mathnet  crossref
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