Estimator for the distribution of the numbers of runs in a random sequence controlled by stationary Markov chain

The sequences of random characters from a finite set $\mathcal A$ with polynomial distributions controlled by a stationary finite-state Markov chain are considered. For numbers of character runs in them, the asymptotic properties of joint distributions are studied. We deduce an estimate for the total variation distance $\rho_{TV}$ between the distribution of a random vector $\varsigma_\mathcal A$ with components being numbers of runs in a controlled sequence of an enough length $T$ and accompanying multidimensional Poisson distribution $\mathrm{Pois}(\lambda_\mathcal A)$. The estimate is $\rho_{TV}\left(\mathcal L(\varsigma_\mathcal A),\mathrm{Pois}(\lambda_\mathcal A)\right)\leq\gamma\left(\gamma T(p^*)^{s_*}+1\right)$, where $\gamma^2=|\mathcal A|^2(2s^*+3)(p^*)^{s_*}$, $s_*$ ($s^*$) is the minimum (maximum) length of run in the set of components of the vector $\varsigma_\mathcal A$, and $p^*$ is the maximum character probability in distributions given on $\mathcal A$. For deriving this estimate, we use the functional variant of Chen–Stein method and an estimation for the total variation distance between the mixed and ordinal Poisson distributions. This estimation is a function of the variance of mixing parameter of mixed Poisson distribution. Using the derived estimate for the total variation distance $\rho_{TV}$, we deduce the multidimensional Poisson and normal limit theorems for the random vector $\varsigma_\mathcal A$ under appropriate conditions for scheme parameters.