Geodesic
About the rate of normal approximation for~the~distribution of the number of repetitions in~a~stationary discrete random sequence
Prikladnaâ diskretnaâ matematika, no. 4 (2022), pp. 15-21

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The paper presents the problem of asymptotic normality of the number of $r$-fold repetitions of characters in a segment of a (strictly) stationary discrete random sequence on the set $\{1,2,\ldots,N\}$ with the uniformly strong mixing property. It is shown that in the case when the uniformly strong mixing coefficient $\varphi(t)$ for an arbitrarily given $\alpha> 0$ decreases as $t^{-6-\alpha}$, then the distance in the uniform metric between the distribution function of the number of repetitions and the distribution function of the standard normal law decreases at a rate of $O(n^{-\delta})$ with increasing sequence length $n$ for any $\delta \in (0;\alpha (32+4\alpha)^{-1 }))$.
Keywords: normal approximation, number of multiple repetitions, stationary random sequence, uniformly strong mixing, distance in uniform metric. 
@article{PDM_2022_4_a1,
     author = {V. G. Mikhailov and N. M. Mezhennaya},
     title = {About the rate of normal approximation for~the~distribution of the number of repetitions in~a~stationary discrete random sequence},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {15--21},
     publisher = {mathdoc},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2022_4_a1/}
}
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V. G. Mikhailov; N. M. Mezhennaya. About the rate of normal approximation for~the~distribution of the number of repetitions in~a~stationary discrete random sequence. Prikladnaâ diskretnaâ matematika, no. 4 (2022), pp. 15-21. http://geodesic.mathdoc.fr/item/PDM_2022_4_a1/