About the rate of normal approximation for the distribution of the number of repetitions in a stationary discrete random sequence

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 $\delta \in (0;\alpha (32+4\alpha)^{-1 }))$.