Let $X_n$ be a DTMC, with transition matrix P and state-space I. Let $Y_m=X_{T_m}$ for $m \in \mathbb{N}$.
Define $T_0=\inf\{n\geq0:X_n\in J\subset I\}$ and $T_{m+1}=\inf\{n> T_{m}:X_n\in J\subset I\}$. These are stopping times, so we can use the strong markov property when conditioning on them. I'm having a bit of trouble understanding a passage from a book.
It goes like this: For $i_0,...,i_{m+1} \in J$ we have $$P(Y_{m+1}=i_{m+1}|Y_0=i_0, ..., Y_m=i_m)=P(X_{T_{m+1}}=i_{m+1}|X_{T_{0}}=i_0,..., X_{T_{m}}=i_m)=P_{i_m}(X_{T_{1}}=i_{m+1})=h^{i_{m+1}}_{i_m}$$ where $h^{i_{m+1}}_{i_m}=P(\inf\{n\geq0:X_n=i_{m+1}\}<\infty|X_0=i_m)$
I don't understand the last two equalities...
I would think that $P(X_{T_{m+1}}=i_{m+1}|X_{T_{m}}=i_m)=P(X_{T_{m+1}-T_m}=i_{m+1}|X_{0}=i_m)$.
Any help would be appreciated.
\begin{align} P(X_{T_{m+1}}=i_{m+1}|X_{T_{0}}=i_0,..., X_{T_{m}}=i_m) &= P(X_{T_{m+1}}=i_{m+1}|X_{T_{m}}=i_m) \\ & \qquad\qquad\qquad\text{it might help to expand $T_{m+1}$ here:} \\ &= P(X_{\inf\{n> T_{m}:X_n\in J\subset I\}}=i_{m+1}|X_{T_{m}}=i_m) \\ &= P(X_{\inf\{n> T_{0}:X_n\in J\subset I\}}=i_{m+1}|X_{T_{0}}=i_m) \\ &= P(X_{T_{1}}=i_{m+1}|X_{T_{0}}=i_m) \\ &= P_{i_m}(X_{T_{1}}=i_{m+1}) \\ &= h^{i_{m+1}}_{i_m}. \end{align}
The probability $P(X_{T_{m+1}-T_m}=i_{m+1}|X_{0}=i_m)$ isn't right because $X_{T_{m+1}-T_m}$ is interpreted as: take the difference of the stopping times $T_{m+1}$ and $T_m$, then advance that number of steps starting at the beginning (step $0$) and take the value of $X$ at that point. The value of $X_{T_{m+1}-T_m}$ therefore could be anything in the state space, not necessarily in set $J$.