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 ρ≥1 of a probability measure μ on the real line with finite moment of order ρ by the empirical measure of N deterministic points. The minimal error converges to 0 as N→+∞ and we try to characterize the order associated with this convergence. In Xu and Berger (2019), Xu and Berger show that, apart when μ 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 μ. They also prove that the order is not smaller than 1/ρ when the support of μ 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/ρ,1, the support has to be bounded and by stating a necessary and sufficient condition in terms of the tails of μ for the order to be equal to some given value in the interval 0,1/ρ, 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/ρ. 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 μ which exhibits the same order of convergence.