I am studying Markov chain. I have a question after reading the first chapter.
We consider a Markov chain $(X, P)$ on a finite state space, irreducible $X$ with transition matrix $P:=(P(\mathcal{x}, y))_{\boldsymbol{x}, y \in X}$; for $n \in \mathbb{N}$, we set $P^n:=\left(P^n(x, y)\right)_{x, y \in X}$. We know that $x$ is communicated with y if there exists $n_0 \in \mathbb{N}$ such that $P^{n_0}>0$.
My question: Does there exist necessarily an $n \ge 1$ such that $P^n(x,y) >0$ for all $x,y \in X$ (If the answer is positive, give a proof, otherwise give the counterexample)?
No, that would require the Markov chain to be aperiodic as well as irreducible. For example, consider the Markov chain with transition matrix $$ P = \pmatrix{0 & 1\cr 1 & 0\cr}$$ where all powers of $P$ are either $P$ or the identity matrix.