Coupling (probability)

In probability theory, coupling is a proof technique that allows one to compare two unrelated random variables(distributions) ${\displaystyle X}$ and ${\displaystyle Y}$ by creating a random vector ${\displaystyle W}$ whose marginal distributions correspond to ${\displaystyle X}$ and ${\displaystyle Y}$ respectively. The choice of ${\displaystyle W}$ is generally not unique, and the whole idea of "coupling" is about making such a choice so that ${\displaystyle X}$ and ${\displaystyle Y}$ can be related in a certain way we desire.

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