Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 215 / 1 / a / Solution

A stopping time $T$ is a strong stationary time iffor every $x,y,t$. Thus $X_T\sim \pi$ and is independent of $T$. The separation distance isFor any strong stationary time,Hence $1-P^t(x,y)/\pi (y)\le {\mathbb{P}}_x(T>t)$, and maximizing proves the claim.