Poisson approximation for the number of non-decreasing runs in Markov chains

Let a sequence $X_1, X_2, \dots, X_n$ be a segment of a stationary irreducible and aperiodic Markov chain with state space $\mathcal{A} = \{1,\dots, N\}$, $N \geqslant 2$. We study the non-overlapping appearances of non-decreasing runs in the sequence $X_1, X_2, \dots, X_n$. By means of Stein method we estimate the total variation distance between the distribution of the number of non-overlapping appearances of non-decreasing monotone runs and the Poisson distribution. As a corollary we prove corresponding limit theorem.