= Coupling inequality for total variation
{title2=$d_{\mathrm{TV}}(\mathcal L(X_t),\mathcal L(Y_t))\leq\mathbb P(T>t)$}
For any <coupling> of <random variables>, the <coupling characterization of total variation distance> gives $d_{\mathrm{TV}}(\mathcal L(X),\mathcal L(Y))\leq\mathbb P(X\ne Y)$, where $\mathcal L$ denotes the <law of a random variable>. If two coupled <Markov chains> stay identical after their coalescence time $T$, then $\{X_t\ne Y_t\}\subseteq\{T>t\}$ and
$$
\lVert\mathcal L(X_t)-\mathcal L(Y_t)\rVert_{\mathrm{TV}}\leq\mathbb P(T>t).
$$
The persistent agreement is essential: a first meeting time does not give this bound if the chains can separate again. A <coalescing coupling> turns a difficult comparison of <probability distributions> into a concrete question about how long a joint simulation takes to merge.
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