Stationary Distribution - Yousef's Notes

The stationary distribution of a Markov Chain with transition matrix $Q$ is the vector $\pi$ for which $$ \pi Q = \pi $$

It can be interpreted as the proportion of times we would be in a specific state if we let the Markov Chain run for a very long time.

Consider an irreducible Markov chain with a finite number of states. Then:

the stationary distribution exists and is unique.
$\pi_i = 1/r_i$, where $r_i$ is the expected amount of time it takes to return to state $i$.

Consider an irreducible and aperiodic Markov chain with a finite number of states (such a chain is usually called ergodic). Then any row of $Q^n$ is the unique stationary distribution, for $n$ large enough.