= Countable-state Markov chain convergence theorem
{title2=$P^n(i,j)\to\pi_j$}
For an <irreducible Markov chain> on a finite or countable state space, a <positive recurrent state> implies a unique <stationary distribution> $\pi$. If the chain is also an <aperiodic Markov chain>, its transition probabilities satisfy the displayed limit for every pair of states. The same limit holds from any initial probability distribution by bounded convergence. Positive recurrence cannot be omitted in the countably infinite case, and aperiodicity is required for ordinary, rather than averaged, convergence. This is the standard discrete-time convergence theorem; the finite-state statement alone does not cover unbounded renewal countdowns.
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