Let $\{X_n\}$ be a Markov Chain with finite state space $S$. Let $T_n$ be the $n$-th hitting time of $A \subset S$ i.e. $n$-th time it hits some state from the set $A$. Is $\{X_{T_n}\}$ a Markov chain ? I don't think so as the usual proof method does not go through (I have checked that). How do I give a more rigorous argument ? Hope the question is clear.
So, $T_n=k$ implies that $X_k\in A$ and of all the $\{X_i: i \leq k-1\}$ exactly $n-1$ random variable lies in $A$