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Suppose {$X_n$} $n = 0, 1, ...$ is an irreducible (Time Homogenous) Discrete Time Markov Chain with transition matrix $P$ such that $P = P^2$. Prove that the chain is aperiodic, positive recurrent, and time reversible if it starts from its stationary distribution.

I've gotten the aperiodic part; essentially as the chain is irreducible, there exists $n > 0$ such that $P_{ii}^n > 0$ and as it is idempotent, we also have $P_{ii} > 0$, thus proving the chain is aperiodic. How do I go about showing it is positive recurrent?

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It suffices to show that the chain has a stationary distribution. Say $v$ is some row vector (to be identified) and set $\pi := vP$. Then $$ \pi P = vP^2 = vP = \pi. $$ So if you can find $v$ so that the elements of $vP$ sum to one (which is required for $\pi$ to be a probability distribution), you are done.

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