On the number of characters matchings in discrete random sequence controlled by Markov chain

The present paper is devoted to studying the properties of the number of pairs of characters matchings in discrete random sequence controlled by Markov chain with finite number of states. Alongside with the random variable we also consider the number of characters which appears in a random sequence exactly $k$ times, $k=0,1,2$. We derive the estimators for total variation distance between distributions of the considered random variables and Poisson distribution. This results allow to develop Poisson and normal limit theorems.