On the limit distribution of a number of runs in polynomial sequence controlled by Markov chain

The present paper is devoted to studying the asymptotic properties of a number of runs in the sequence of discrete random variables controlled by Markov chain with a finite number of states. A chain state at each step determines the law of characters distribution in the controlled sequence at this step. This random sequence represents a model of hidden Markov chain. Using Chen–Stein method we estimate the total variation distance between the distribution of the number of runs with length not less than predetermined length in the random sequence controlled by Markov chain and the accompanying Poisson distribution. For this purpose we first consider the sequence of independent inhomogeneous polynomial random variables, and then we use an approach which allows to get the estimate for total variation distance between mixed Poisson distribution and Poisson distribution with the parameter which equals to an average number of runs with length not less than predetermined. The estimate is based on both the variance of the mixed Poisson distribution parameter and the estimate obtained earlier for the total variation distance for the polynomial scheme. Separately we consider the case of a stationary Markov chain. Using derived estimates we investigate Poisson and normal limit theorems for the number of runs with length not less than predetermined, as well as the limit distribution for the maximal run length in a controlled sequence.