Strong stationary time

A randomized stopping time $\tau$ is a strong stationary time for a Markov chain with stationary distribution $\pi$ when $X_{\tau }$ has distribution $\pi$ and is independent of $\tau$. Equivalently,

For a finite Markov chain with stationary distribution $\pi$, the separation distance from an initial state $x$ isIf $\tau$ is a strong stationary time, then $s_x(t)\le {\mathbb{P}}_x(\tau >t)$, and total variation distance is at most separation distance.