Equilibrium distribution

Question

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?

Answer 1

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.