Estimates of accuracy of the Poisson approximation for the distribution of number of runs of long string repetitions in a Markov chain

Let $X_0,X_1,\ldots$ be a simple ergodic Markov chain with a finite set of states and $\tilde\xi_{n,k}(s)$ be a number of runs of $k$-fold repetitions of strings having length $s$. Estimates of accuracy of the Poisson approximation for the distribution of $\xi_{n,k}(s)$ in the sequence $X_0,X_1,\ldots,X_{n+s-1}$ are obtained, these estimates are uniform over $k$. \def\acknowledgementname{Funding