= One-dimensional quadratic transport uniqueness criterion
{title2=$\pi\{x\le a,y\le b\}=\min\{F(a),G(b)\}$}
For atomless source and target probabilities with continuous strictly increasing cumulative distributions, the squared-distance optimal coupling is unique. Two-point optimality forbids crossed support pairs. This makes every pair of source and target lower-tail events nested up to null sets and forces the displayed joint distribution, which is the common-quantile coupling and is induced by $G^{-1}F$.
Back to article page
