The rate of normal approximation for the distribution of the number of multiple repetitions of characters in a stationary random sequence

We study the asymptotic normality of the number of $r$-fold characters repetitions in a segment of length $n$ of a strictly stationary random sequence with values in a finite set that satisfies the uniformly strong mixing condition. It is shown that if there exists a number $\alpha> 0$ such that the uniformly strong mixing coefficient $\varphi(t)$ decreases as $t^{-6-\alpha}$, then the distance in the uniform metric between the distribution function of the standardized number of repetitions of multiplicity $r$ and the distribution function of the standard normal law decreases at a rate of $O(n^{-\delta})$ for any $\delta \in (0,\alpha (32+4\alpha)^{ -1})$ with increasing of segment length $n$.