On the rate of convergence of empirical measure in $\infty-$Wasserstein distance for unbounded density function

TL;DR

This paper derives a new rate of convergence for the $ abla$-Wasserstein distance between empirical and true measures in one dimension, specifically addressing cases with unbounded densities, extending prior results.

Contribution

It extends existing convergence results of the $ abla$-Wasserstein distance to distributions with unbounded densities in one dimension.

Findings

Established a new convergence rate for unbounded densities

Extended previous results by Trilllos and Slepčev

Applicable to one-dimensional absolutely continuous measures

Abstract

We consider a sequence of identically independently distributed random samples from an absolutely continuous probability measure in one dimension with unbounded density. We establish a new rate of convergence of the $\infty-$Wasserstein distance between the empirical measure of the samples and the true distribution, which extends the previous convergence result by Trilllos and Slep\v{c}ev to the case that the true distribution has an unbounded density.