Measuring association with Wasserstein distances

Let n ??? ??(??,v) be a coupling between two probability measures ?? and v on a Polish space. In this article we propose and study a class of nonparametric measures of association between ?? and v, which we call Wasserstein correlation coefficients. These coefficients are based on the Wasserstein distance between v and the disintegration nx1 of n with respect to the first coordinate. We also establish basic statistical properties of this new class of measures: we develop a statistical theory for strongly consistent estimators and determine their convergence rate in the case of compactly supported measures ?? and v. Throughout our analysis we make use of the so-called adapted/bicausal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglb??ck, Wiesel. Estimating processes in adapted Wasserstein distance. 2020]. Our approach applies to probability laws on general Polish spaces.