This question is from Resnick. It is not a homework question, I am merely preparing for my exam by doing practice problems.
Question: Let $\{X_n\}$ be a Markov chain with state space $S$, and let $T$ be a stopping time so that the event $[T=n]$ is determined by $X_0,....X_n$. Suppose for all $i \in S$ that $P_{i}[T<\infty]=1$. Suppose that the state space is $S^{\infty}=\{(s_0,s_1,........): s_i \in S, i=0,1,.....\}$. Define $T_1=T$ and for $n \geq 2$ define $$T_n(s_0,s_1,.....)=T(s_{T_{n-1}+1},s_{T_{n-1}+2},......)$$ so that $T_n$ is $T$ applied to the segment of $(s_0,s_1,....)$ beyond index $T_{n-1}$. Show that $\{X_{T_n}, n \geq 1\}$ is a Markov Chain.
Thoughts: I was having trouble with this question since I think I should use the Strong Markov Property since $T$ is a stopping time. So I think I need to show somehow that: \begin{align*} &P(X_{T_{n+1}}=i_{n+1} \mid X_{T_1}=i_1,....X_{T_n}=i_n) \\ &=P(X_{T_{n+1}}=i_{n+1} \mid X_{T_{n}}=i_n)\\ &=P(X_{T_{1}}=i_{n+1} \mid X_{T_{0}}=i_n)\\ \end{align*} Something like this, not exactly sure what I'm doing. The problem is I don't really understand the question. I don't understand "define $$T_n(s_0,s_1,.....)=T(s_{T_{n-1}+1},s_{T_{n-1}+2},......)$$ so that $T_n$ is $T$ applied to the segment of $(s_0,s_1,....)$ beyond index $T_{n-1}$".
I don't really understand the meaning of $T_n$ and how to use this to show that $X_{T_n}$ is a Markov chain. I'm guessing $T_n$ is a stopping time. Sorry for all my confusion. If anyone could help me get started that would be much appreciated. Thanks.