Let $(X_n)_{n\in\mathbb{N}_0}$ be a homogeneous markov chain with starting distribution $\mu$, transition matrix $P$ and $P(x,x)<1$ for all $x\in S$, and
$\tau_0:=0$ and $\tau_{k+1}:=\inf\{n\geq \tau_k:X_n \neq X_{\tau k}\}$ for all $k\in \mathbb{N}_0$
Prove with the strong markov property that $Y_k:=X_{\tau k}$ for all $k\in\mathbb{N}_0$ is a homogeneous markov chain. How are the starting distribution and the transition matrix of $Y_k$ defined?
Any help is appreciated.