On the rate of convergence of empirical measure in ∞-Wasserstein distance for unbounded density function

We consider a sequence of identical 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 ∞ ∞ -Wasserstein distance between the empirical measure of the samples and the true distribution, which extends the previous convergence result by Trillos and Slepčev to the case that the true distribution has an unbounded density.