# Does product Markov chain hit diagonal?

Suppose $(X_n)$ is a Markov chain (discrete time countable state space $S$) with irreducible positive recurrent transition matrix $P$. Consider the product chain $(Y_n,Z_n)$ given by choosing $(Y_n)$ and $(Z_n)$ independent Markov chains with transition matrix $P$ with arbitrary initial conditions $Y_0, Z_0$. I know that if $P$ is aperiodic, then the product chain will a.s. hit the diagonal $D=\{(x,x): x \in S\}$ because the product chain is itself an ergodic Markov chain. If $P$ instead has period $k>1$, is this result still true?

Consider $P$ given by the deterministic process $0 \to 1 \to 0 \to \dots,Y_0=0,Z_0=1$ to get a counterexample.
• What do you mean by the distributions being independent? Note that if $P(Y_0=0,Z_0=1)=1$ then $Y_0$ and $Z_0$ are independent. – Did Oct 19 '17 at 18:38