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We are interested in the approximation in Wasserstein distance with index$\rho\ge 1$ of a probability measure $\mu$ on the real line with finite momentof order $\rho$ by the empirical measure of $N$ deterministic points. Theminimal error converges to $0$ as $N\to+\infty$ and we try to characterize theorder associated with this convergence. In \cite{xuberger}, Xu and Berger showthat, apart when $\mu$ is a Dirac mass and the error vanishes, the order is notlarger than $1$ and give a sufficient condition for the order to be equal tothis threshold $1$ in terms of the density of the absolutely continuous withrespect to the Lebesgue measure part of $\mu$. They also prove that the orderis not smaller than $1/\rho$ when the support of $\mu$ is bounded and notlarger when the support is not an interval. We complement these results bychecking that for the order to lie in the interval $\left(1/\rho,1\right)$, thesupport has to be bounded and by stating a necessary and sufficient conditionin terms of the tails of $\mu$ for the order to be equal to some given value inthe interval $\left(0,1/\rho\right)$, thus precising the sufficient conditionin terms of moments given in \cite{xuberger}. In view of practical application,we emphasize that in the proof of each result about the order of convergence ofthe minimal error, we exhibit a choice of points explicit in terms of thequantile function of $\mu$ which exhibits the same order of convergence.

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