Coalescing coupling
= Coalescing coupling
{title2=$T$}
A coalescing <coupling of probability distributions> of two <Markov chains> keeps them identical after a meeting time $T$. The <coupling inequality for total variation> gives $\|P^t(x,\cdot)-P^t(y,\cdot)\|_{\mathrm{TV}}\leq\mathbb P(T>t)$. A uniform bound on $\mathbb E T$, combined with <Markov inequality>, bounds the <mixing time of a Markov chain>.