Optimal Transportation Plans and Convergence in Distribution

Explicit expression of mappings optimal transportation plans for the Wasserstein distance in Rp,p>1, are not generally available. Therefore, it is of great interest to provide results which justify the practical use of simulation techniques to obtain approximate optimal transportation plans. This is done in this paper, where we obtain the consistency of the empirical optimal transportation plans. Our results can also be employed to justify a definition of multidimensional complete dependence.