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

Anning Liu; Jian-Guo Liu; Yulong Lu

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

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arxiv 1807.08365 v2 pith:WBSAXYKJ submitted 2018-07-22 math.PR math.STstat.TH

Anning Liu , Jian-Guo Liu , Yulong Lu This is my paper · ORCID

classification math.PRmath.STstat.TH

keywords convergencedensitymeasureunboundeddistancedistributionempiricalinfty-

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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.

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