Coupling

Summary

A coupling of two probability measures $\mu$ and $\nu$ is a joint law of a pair $(X, Y)$. In it, $X$ has law $\mu$ and $Y$ has law $\nu$. The two marginals are fixed, but the dependence between $X$ and $Y$ is free to choose. A clever choice turns a comparison of two laws into a comparison of two random variables on one space. Couplings are a main tool for bounds in total variation distance, for convergence of Markov chains, for stochastic ordering and in optimal transport.