 Solution

Condition on the intermediate state at time $m$. The Markov property and time homogeneity giveThese nonnegative sums are valid on a countable state space. Starting from $P(0)=I$, $P(1)=P$, induction gives $\boxed{P(n)=P^n}$. Conditioning on the initial state now yields $\boxed{\mathrm{Pr}(X_n=i)=(\mu P^n{)}_i}$. If $\mu P=\mu$, induction gives $\mu P^n=\mu$ for every $n$, so all one-time distributions equal the stationary initial distribution.