PUB - Publikationen an der Universität Bielefeld

The aim of this article is to write the p-Wasserstein metric $ W_p $ Wp with the p-norm, $ p\in [1,\infty ) $ p is an element of[1,infinity), on $ \mathbb {R}<^>d $ Rd in terms of copula. In particular for the case of one-dimensional distributions, we get that the copula employed to get the optimal coupling of the Wasserstein distances is the comotonicity copula. We obtain the equivalent result also for d-dimensional distributions under the sufficient and necessary condition that these have the same dependence structure of their one-dimensional marginals, i.e that the d-dimensional distributions share the same copula. Assuming $ p\neq q $ p not equal q, p,q $ \in [1,\infty ) $ is an element of[1,infinity) and that the probability measures mu and nu are sharing the same copula, we also analyze the Wasserstein distance $ W_{p,q} $ Wp,q discussed in [Alfonsi and Jourdain. A remark on the optimal transport between two probability measures sharing the same copula. Statist. Probab. Lett. 84 (2014) 131-134.] and get an upper and lower bounds of $ W_{p,q} $ Wp,q in terms of $ W_p $ Wp, written in terms of comonotonicity copula. We show that as a consequence the lower and upper bound of $ W_{p,q} $ Wp,q can be written in terms of generalized inverse functions.