Copulas farthest from independence in quadratic Wasserstein distance

Source abstract

Let $Π$ denote the independence copula and $M,W$ the upper and lower Fréchet--Hoeffding copulas. Catalano and Lavenant (2025) conjectured that $M$ and $W$ maximize the quadratic Wasserstein distance from $Π$ among all bivariate copulas. We prove this conjecture and characterize all equality cases: $\mathcal W_2^2(C,Π)\leq 1/10$, with equality if and only if $C\in\{M,W\}$. We also determine explicitly the optimal Monge map from $Π$ to $M$. The proof is constructive and combines the optimal transport from independence to the diagonal, a sharp convex-order inequality for 1-Lipschitz functions, the conditional convex order, and a coupling construction based on conditional comonotonicity and the supermodular order. As a consequence, we obtain a normalized Wasserstein-based dependence measure that characterizes independence and attains its maximal value exactly for comonotone and countermonotone dependence.