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

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 $\mathrm{\infty }-$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.