Coupling inequality for total variation ($d_{\mathrm{TV}}(\mathcal{L}(X_t),\mathcal{L}(Y_t))\le \mathbb{P}(T>t)$)

For any coupling of random variables, the coupling characterization of total variation distance gives $d_{\mathrm{TV}}(\mathcal{L}(X),\mathcal{L}(Y))\le \mathbb{P}(X\ne Y)$, where $\mathcal{L}$ denotes the law of a random variable. If two coupled Markov chains stay identical after their coalescence time $T$, then $\{X_t\ne Y_t\}\subseteq \{T>t\}$ andThe 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.