Continuous-time Markov chain convergence theorem (${\lim }_{t\to \infty }{P}_t(i,j)={\pi }_j$)

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 $t=nh+r$, $0\le r<h$, applying $P_r$ contracts total variation distance and preserves $\pi$, giving the full continuous-time limit.