The Wasserstein distance and approximation theorems

Abstract

By an extension of the idea of the multivariate quantile transform we obtain an explicit formula for the Wasserstein distance between multivariate distributions in certain cases. For the general case we use a modification of the definition of the Wasserstein distance and determine optimal ‘markov-constructions’. We give some applications to the problem of approximation of stochastic processes by simpler ones, as e.g. weakly dependent processes by independent sequences and, finally, determine the optimal martingale approximation to a given sequence of random variables; the Doob decomposition gives only the ‘one-step optimal’ approximation.