 Pairwise mixing diameter

For a finite Markov chain with transition matrix $P$, define $\bar{d}(t)={\max }_{x,y}\Vert P^t(x,\cdot )-P^t(y,\cdot ){\Vert }_{\mathrm{TV}}$. If $\pi$ is a stationary distribution and $d(t)={\max }_x\Vert P^t(x,\cdot )-\pi {\Vert }_{\mathrm{TV}}$, then $d(t)\le \bar{d}(t)\le 2d(t)$. The first bound follows by writing $\pi ={\sum }_y\pi (y)P^t(y,\cdot )$ and using the triangle inequality; the second uses the triangle inequality through $\pi$.