= Continuous-time Markov chain convergence theorem
{title2=$\lim_{t\to\infty}P_t(i,j)=\pi_j$}
A countable <nonexplosive continuous-time Markov chain> which is an <irreducible Markov chain> and has an <invariant distribution> $\pi$ is <positive recurrent>; its <transition probabilities> tend to $\pi$ from every initial <Markov chain state>.
To see why continuous time removes periodicity, fix $h>0$. Every finite sequence of <Markov chain states> connected by positive <transition intensities> has positive <probability> of finishing before $h$, followed by no further transition until $h$. Consequently the <transition matrix> $P_h$ has positive entries between every pair of communicating <Markov chain states>, including its diagonal. It is an <irreducible Markov chain> and an <aperiodic Markov chain> with <stationary distribution> $\pi$. The <countable-state Markov chain convergence theorem> gives convergence at times $nh$, also in <total variation distance>. For $t=nh+r$, $0\leq r<h$, applying $P_r$ contracts <total variation distance> and preserves $\pi$, giving the full continuous-time limit.
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