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

TL;DR

This paper investigates the rate at which empirical measures approximate a probability measure on the real line in Wasserstein distance, providing conditions for different convergence orders based on the measure's support and tail behavior.

Contribution

It extends previous work by characterizing the convergence order in Wasserstein distance, especially for orders between 0 and 1, with explicit point choices related to the measure's quantile function.

Findings

Convergence order is at most 1, with conditions for equality.

Order in (1/ρ, 1) requires bounded support.

Necessary and sufficient tail conditions for order in (0, 1/ρ).

Abstract

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 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. In \cite{xuberger}, 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 $\left(1/\rho,1\right)$, 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 $\left(0,1/\rho\right)$, thus precising the sufficient condition in 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 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.