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 Diskr. Mat., 2011, Volume 23, Issue 2, Pages 41–52 (Mi dm1139)

Asymptotic normality of the number of values of $m$-dependent random variables which occur a given number of times

V. P. Chistyakov

Abstract: We consider a stationary sequence $x_1,x_2,…,x_t,…$ of random variables each of which takes values in the set $\mathcal N=\{1,…,N\}$. Let $\mu_r=\mu_r(n,N)$ be the number of values in the set $\mathcal N$ which occur $r$ times in the first $n$ elements of the sequence. In this paper, we obtain asymptotic formulas for the mean value $\mathbf E\mu_r(n,N)$ and the variance $\mathbf D\mu_r(n,N)$ and give sufficient conditions for asymptotic normality of the random variables $\mu_r(n,N)$ as $n,N\to\infty$ under the condition that the sequence consists of $m$-dependent random variables and the parameters of the distribution of the sequence vary in the so-called central domain.

DOI: https://doi.org/10.4213/dm1139

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English version:
Discrete Mathematics and Applications, 2011, 21:1, 23–37

Bibliographic databases:

Document Type: Article
UDC: 519.24

Citation: V. P. Chistyakov, “Asymptotic normality of the number of values of $m$-dependent random variables which occur a given number of times”, Diskr. Mat., 23:2 (2011), 41–52; Discrete Math. Appl., 21:1 (2011), 23–37

Citation in format AMSBIB
\Bibitem{Chi11} \by V.~P.~Chistyakov \paper Asymptotic normality of the number of values of $m$-dependent random variables which occur a~given number of times \jour Diskr. Mat. \yr 2011 \vol 23 \issue 2 \pages 41--52 \mathnet{http://mi.mathnet.ru/dm1139} \crossref{https://doi.org/10.4213/dm1139} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2865905} \elib{http://elibrary.ru/item.asp?id=20730382} \transl \jour Discrete Math. Appl. \yr 2011 \vol 21 \issue 1 \pages 23--37 \crossref{https://doi.org/10.1515/DMA.2011.002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79953301814}