Geodesic
Asymptotic normality of the number of values of $m$-dependent random variables which occur a~given number of times
Diskretnaya Matematika, Tome 23 (2011) no. 2, pp. 41-52

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We consider a stationary sequence $x_1,x_2,\dots,x_t,\dots$ of random variables each of which takes values in the set $\mathcal N=\{1,\dots,N\}$. 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. 
@article{DM_2011_23_2_a2,
     author = {V. P. Chistyakov},
     title = {Asymptotic normality of the number of values of $m$-dependent random variables which occur a~given number of times},
     journal = {Diskretnaya Matematika},
     pages = {41--52},
     publisher = {mathdoc},
     volume = {23},
     number = {2},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2011_23_2_a2/}
}
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V. P. Chistyakov. Asymptotic normality of the number of values of $m$-dependent random variables which occur a~given number of times. Diskretnaya Matematika, Tome 23 (2011) no. 2, pp. 41-52. http://geodesic.mathdoc.fr/item/DM_2011_23_2_a2/