We have an irreducible Markov chain with a not necessarily finite state space. It has a transition matrix $P$ such that $P^2=P$. Prove (1) the chain is aperiodic, and (2) prove $p_{ij}=p_{jj}$ $\forall i,j \in S$. Then, find a stationary distribution in terms of $P$.
I'm fairly certain I can solve (1). Here is my attempt:
Since $P^2=P$, $P^3=P^2P = PP=P$. By induction, it's obvious that $P^n=P$ $\forall n\geq 0$. Since chain is irreducible, take $i \in S$ state space $\Rightarrow $ $p_{ii}^n >0$. Also, $p_{ii}^n = p_{ii}$. Thus, we can return in one step. So the chain is aperiodic.
For part (2), I don't understand how to complete the proof entirely, but I think it might be useful to use a similar statement that $p_{ij}^n=p_{ij}>0$ $\forall i,j \in S$. I think I need to show that it is positive recurrent, but how?
So, in short, did I do part 1 correctly, and can you finish/guide me in part 2?
(1) looks ok for me, too. As for (2), there is 3 possibility for an irreducible aperiodic Markov chain: it either transient, or null recurrent, or positive recurrent. In our case, the chain can not be transient, because for a transient chain we would have for all $i \in S$
$$ \sum\limits _{n=1} ^\infty p^{(n)} _{ii} < \infty, $$
nor can it be null recurrent, since for a null recurrent chain we would have
$$ p^{(n)} _{ii} \to 0 \ , \ n \to \infty. $$
Therefore, the chain is positive recurrent, and there exist a stationary distribution $\{\pi _i \}$, and $p^{(n)} _{ji} \to \pi _i$ for all $i,j \in S$. As you noticed, $p^{(n)} _{ji} = p _{ji}$, and the remaining part should not cause any troubles.
All this criteria for transience, null recurrence and positive recurrence may be found in the book by Shiryaev, 'Probability' (but surely, in some other places as well).