Theorem (Dobrushin Convergence Theorem)

For any two probability measures, $\pi,\pi'$ it follows that $$\|\pi P-\pi'P\|\le(1-\delta(P))\|\pi-\pi'\|$$hence, $$\pi_{k+1}=\pi_{k}P\quad(=:T(\pi_{k}))$$converges to the unique invariant probability measure $\pi^{*}$ $$\|\pi_{n}-\pi^*\|\le2(1-\delta(P))^{n}$$