How can block-matrices be irreducible?

Question

Suppose we define two matrices P1 and P2 as follows:

• both are 2x2 matrices
• both have strictly positive entries

And then we define P to be a 4x4 transition matrix of the form

P =
[P1 0]
[0 P2]

It's clear here that we have two communicating classes, i.e. states 1 and 2 communicate, and states 3 and 4 communicate, however there is no communication going on between these two groups. This would imply that this matrix is not irreducible, and therefore does not have a limiting distribution.

I know that my understanding here is incorrect because I know that P actually does have a limiting distribution, but I'm not sure how this can be the case given that we know P to have multiple communicating classes.

Answer 1

This would imply that this matrix is not irreducible, and therefore does not have a limiting distribution.

Well, it does have a limiting distribution, but not a unique limiting distribution:

Let $\pi_1\in\mathbb{R}^2$ be such that $\pi_1 P_1=\pi_1$ (Or $P_1\pi_1=\pi_1$ I always forget what you want in this context) and $\pi_2\in\mathbb{R}^2$ be such that $\pi_2 P_2=\pi_2$, then $\lambda\pi_1\times0+(1-\lambda)0\times\pi_2$ is a limiting distribution of your process $P_1\times P_2$ for any $\lambda\in(0,1)$.