Let $(X_n)_{n\in\mathbb{N_0}}$ be a homogeneous markov chain with initial distribution $\mu$ and transition matrix $P$.
Is $(X_{2n})_{n\in\mathbb{N}_0}$ a homogeneous markov chain? What are the initial distribution and transition matrix of it?
How can I prove it?
Let $p_{ij}$ be the probability that the chain will be in state $j$ at time $t+1$ given that it is in state $i$ at time $t$, so that the transition matrix is $$P=(p_{ij})_{n\times n}$$ Then then probability that the chain will be in state $j$ at time $t+2$ given that it is in state $i$ at time $t$ is $$\sum_{k=1}^np_{ik}p_{kj}\tag1$$ by the law of total probability, because the chain must first transition from state $i$ to some state $k$, and then from state $k$ to state $j$.
I'm sure you recognize the expression in $(1)$. It's just $P^2_{ij}$. That is, the transition matrix is $P^2$. Since this depends only on the state, and not the time, $X_{2n}$ is homogeneous.