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I have learned that irreducible ergodic Markov process has an equilibrium distribution. Does equilibrium distribution exist if the Markov chain is not irreducible? Is the reducible Markov chain reversible?

Yuteng
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Consider a Markov chain with an absorbing state in state $i$. Then, certainly it is not irreducible; but it has an equilibrium distribution, namely $(0, 0, ...1, 0)$ where the $1$ is in the $i^{th}$ position of the probability vector.

thesmallprint
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  • I understand what you mean, thanks a lot. could you please have a look at the second question? – Yuteng May 13 '18 at 17:15