On repetitions of long tuples in a Markov chain

Let $X_0,X_1,\ldots$ be a simple ergodic finite Markov chain. We prove limit theorems for the distribution of the number $\tilde\xi(s,n)$ of events
$$\{X_{i-1}\ne X_{j-1},\ X_{i+k}= X_{j+k},\ k=0,\ldots,s-1\},\quad 1\le i<j\le n,$$
when $s,n\to\infty$. Limit theorems for distributions of some random variables connected with $\tilde\xi(s,n)$ are derived as corollaries.