Copulas farthest from independence in quadratic Wasserstein distance
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: 𝒲22(C,Π)≤ 1/10, with equality if and only if C∈{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.
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BibTeX
@misc{ansari2026copulas,
title = {{Copulas farthest from independence in quadratic Wasserstein distance}},
author = {Jonathan Ansari},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23940},
eprint = {2609.23940},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23940},
}
RIS
TY - UNPB
TI - Copulas farthest from independence in quadratic Wasserstein distance
AU - Jonathan Ansari
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23940
UR - https://arxiv.org/abs/2609.23940
AB - 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: 𝒲_2^2(C,Π)≤ 1/10, with equality if and only if C∈{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.
ER -