Approximation rate in Wasserstein distance of probability measures on the real line by deterministic empirical measures

We are interested in the approximation in Wasserstein distance with index rho >= 1 of a probability measure mu on the real line with finite moment of order rho by the empirical measure of N deterministic points. The minimal error converges to 0 as N -> +infinity and we try to characterize the order associated with this convergence. In Xu and Berger (2019), Xu and Berger show that, apart when mu is a Dirac mass and the error vanishes, the order is not larger than 1 and give a sufficient condition for the order to be equal to this threshold 1 in terms of the density of the absolutely continuous with respect to the Lebesgue measure part of mu. They also prove that the order is not smaller than 1/rho when the support of mu is bounded and not larger when the support is not an interval. We complement these results by checking that for the order to lie in the interval (1/rho,1), the support has to be bounded and by stating a necessary and sufficient condition in terms of the tails of mu for the order to be equal to some given value in the interval (0,1/rho), thus precising the sufficient condition in terms of moments given in Xu and Berger (2019). We also give a necessary condition for the order to be equal to the boundary value 1/rho. In view of practical application, we emphasize that in the proof of each result about the order of convergence of the minimal error, we exhibit a choice of points explicit in terms of the quantile function of mu which exhibits the same order of convergence. (c) 2021 Elsevier Inc. All rights reserved.