发表机构
Department of Mathematics, Paris Lodron University of Salzburg（萨尔茨堡巴黎洛德龙大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了Catalano和Lavenant关于二次Wasserstein距离下最远离独立性的copulas的猜想，即M和W达到最大值1/10，并刻画了等号条件，同时给出最优Monge映射及新的相依性度量。
AI 中文摘要
设$\Pi$表示独立性copula，$M,W$分别为上、下Fréchet--Hoeffding copulas。Catalano和Lavenant（2025）猜想在所有二元copulas中，$M$和$W$最大化与$\Pi$的二次Wasserstein距离。我们证明了这个猜想并刻画了所有等号情形：$\mathcal W_2^2(C,\Pi)\leq 1/10$，等号成立当且仅当$C\in\{M,W\}$。我们还显式确定了从$\Pi$到$M$的最优Monge映射。证明是构造性的，结合了从独立性到对角线的最优传输、1-Lipschitz函数的尖锐凸序不等式、条件凸序，以及基于条件共单调性和超模序的耦合构造。作为推论，我们获得了一个归一化的基于Wasserstein的相依性度量，它刻画独立性，并恰好在共单调和反单调相依时达到最大值。
英文摘要
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.
Comments16 pages, 1 figure
