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

28 pagesWe are interested in the approximation in Wasserstein distance with index $\rho\ge 1$ 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 $N\to+\infty$ and we try to characterize the order associated with this convergence. Apart when $\mu$ is a Dirac mass and the error vanishes, the order is not larger than $1$. We give a necessary condition and 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 $\mu$. We also check that for the order to lie in the interval $\left(1/\rho,1\right)$, the support of $\mu$ has to be a bounded interval, and that, when $\mu$ is compactly supported, the order is not smaller than $1/\rho$. Last, we give 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 $\left(0,1/\rho\right)$