Definition of a a Markov renewal process:
Let the states of a process be denoted by the set $E= \{0,1,2, \dots\}$ and let the transitions of the process occur at epochs $t_0 =0,t_1,t_2, \dots$. Let $X_n$ denote the transition occurring at epoch $t_n$. We say that $\{X_n, t_n\}$ is a Markov renewal process, if $$P(X_{n+1}=k,t_{n+1}-t_n \leq t ~|~ X_0,\dots, X_n,t_0,\dots,t_n) = P(X_{n+1}=k,t_{n+1}-t_n \leq t ~|~X_n).$$
Question
Define $Y_i=X_n$ on $t_n \leq i <t_{n+1}$. Is it true that $$P(Y_{j+1}=k~|~Y_0, \dots ,Y_j)=P(Y_{j+1}=k~|~Y_j,\tau_j),$$ where $\tau_i=\max\{ t ~|~Y_i= Y_k \text{ for all } k \text{ with } i-t\leq k \leq i\}$ are the sojourn times?
