Pairwise mixing diameter
= Pairwise mixing diameter
{title2=$\bar d(t)$}
For a finite <Markov chain> with <transition matrix> $P$, define $\bar d(t)=\max_{x,y}\|P^t(x,\cdot)-P^t(y,\cdot)\|_{\mathrm{TV}}$. If $\pi$ is a <stationary distribution> and $d(t)=\max_x\|P^t(x,\cdot)-\pi\|_{\mathrm{TV}}$, then $d(t)\leq\bar d(t)\leq2d(t)$. 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 $\pi$.