I have a problem with the entropy rate when two ergodic Markov processes who are independent of each other having a non-ergodic joint. More specifically let us consider two finite-state Markov processes $\mathscr{P}_1$ and $\mathscr{P}_2$ with transition matrices $\Pi_1$ and $\Pi_2$, respectively. Let us assume that $\Pi_i$, $i=1, 2$, is a irreducible row stochastic matrix of period $p_i$, and $\gcd(p_1, p_2) = p > 1$. Because $\Pi_i$, $i=1, 2$, is irreducible, we know that $\mathscr{P}_i$ is ergodic.

Assume that $\mathscr{P}_1$ and $\mathscr{P}_2$ are independent, we have their joint process has transition matrix $\Pi =\Pi_1\otimes\Pi_2$, where $\otimes$ means the Kronecker product. However, since $p>1$, $\Pi$ is **not** irreducible. In fact, we can reorder the columns and rows of $\Pi$ simultaneously so that it becomes $\textrm{diag}(A_1, \ldots, A_p)$ where $A_i$, $i=1,\dots, p$, is a irreducible row stochastic matrix.

**My question is:** Let us denote entropy rate of the Markov process given by transition matrix $A$ by $\mathcal{H}(A)$, do we have $$\mathcal{H}(A_i) = \mathcal{H}(\Pi_1) + \mathcal{H}(\Pi_2)$$ for all $i = 1,\dots, p$?

The statement holds for a few examples I tried (some of them are not even Markov process, but were generated by a little bit more complicated models). But I wasn't successful in comping up either a proof or disproof.

Please help! Thank you in advance.