> [!definition] > > Let $\seqi{X}$ be a sequence of [[Random Variable|random variables]] defined on the [[Probability|probability spaces]] $(\Omega_i, \cf_i, \bp_i)$. > > For distributions: A **coupling** of $\seqi{\mu}$ is a collection of random variables defined on a *common* probability space $(\Omega, \cf, \bp)$ such that the distributions $\mu_i = \mu_{Y_i}$ for all $i \in I$. > > For random variables: A **coupling** of $\seqi{X}$ is a collection of random variables defined on a *common* probability space $(\Omega, \cf, \bp)$ such that the [[Cumulative Distribution Function|CDFs]] $F_{Y_i} = F_{X_i}$ for all $i \in I$.