On multiple repetitions of long tuples in a Markov chain

Let $X_0,X_1,\dots$ be a simple ergodic Markov chain with $N$ states and $\tilde\xi_{n,k}^{(m)}(s)$ be the number of $m$-series of $k$-repetitions of $s$-tuples in the chain segment $X_0,X_1,\dots,X_{n+s+m}$. The sufficient conditions for the distribution of the vector $\tilde\Xi_{n,k,M}(s)=(\tilde\xi_{n,k}^{(1)}(s),\dots,\tilde\xi_{n,k}^{(M)}(s))$ to converge to the multidimensional Poisson distribution are found. This permits to prove limit theorems for the distributions of some random variables connected with $\tilde\Xi_{n,k,M}(s)$.