Measuring association with Wasserstein distances

Let ???(?,?) be a coupling between two probability measures ? and ? on Polish space. In this article we propose study class of nonparametric association ?, which call Wasserstein correlation coefficients. These coefficients are based the distance disintegration ?x1 ? with respect to first coordinate. We also establish basic statistical properties new measures: develop theory for strongly consistent estimators determine their convergence rate in case compactly supported ?. Throughout our analysis make use so-called adapted/bicausal distance, particular rely results established [Backhoff, Bartl, Beiglböck, Wiesel. Estimating processes adapted distance. 2020]. Our approach applies laws general spaces.