Markov chain: infinite number of stationary distributions

Question

Is it possible to construct a Markov chain having an infinite number of stationary distributions $\pi_i$? Maybe also with a finite set of states $S$?

Maybe someone can explain why the following Markov chain has an infinite number of stationary distrbutions (see comments) instead of two stationary distrbutions like $[1, 0]$ and $[0, 1]$.

Yes, take any Markov chain with two irreducible classes of states. — Michael, Apr 07 '19 at 00:44
... which can be as small as two absorbing states and nothing else — Henry, Apr 07 '19 at 00:48
Why does a Markov chain with two irreducible classes have an infinite number of stationary distributions - and not just two? — ScientiaEtVeritas, Apr 07 '19 at 09:28
You can take any (infinite number of) weighted average of the two stationary distributions to get another stationary distributions. — Minus One-Twelfth, Apr 07 '19 at 10:33

score 0 · Answer 1 · answered Apr 07 '19 at 10:13

Check the following, $[1,0]$ is a stationary distribution, $[0,1]$ is also a stationary distribution.

Now, consider their convex combination, that is

$$[\lambda, 1-\lambda]$$ where $\lambda \in [0,1]$ is also a stationary distribtuion. Since there are infinitely many choices for $\lambda$, we have infinite number of stationary distribution.

Markov chain: infinite number of stationary distributions

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