Coalescing coupling ($T$)

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 $\Vert P^t(x,\cdot )-P^t(y,\cdot ){\Vert }_{\mathrm{TV}}\le \mathbb{P}(T>t)$. A uniform bound on $\mathbb{E}T$, combined with Markov inequality, bounds the mixing time of a Markov chain.

Suppose an involution $R$ of a Markov chain satisfies $P(Rx,Ry)=P(x,y)$. Start one chain at $x$, set the other equal to its image under $R$ until the first hitting time of the fixed-point set of $R$, and subsequently use identical transitions. This is a coalescing coupling, because the symmetry preserves each marginal transition matrix and both chains coincide at that hitting time.