On the rate of convergence of empirical measure in $\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 $\infty-$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 of unbounded density.

[1]  A E Bostwick,et al.  THE THEORY OF PROBABILITIES. , 1896, Science.

[2]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[3]  J. Kiefer,et al.  Asymptotic Minimax Character of the Sample Distribution Function and of the Classical Multinomial Estimator , 1956 .

[4]  R. Dudley.  The Speed of Mean Glivenko-Cantelli Convergence , 1969 .

[5]  János Komlós,et al.  On optimal matchings , 1984, Comb..

[6]  Frank Thomson Leighton,et al.  Tight bounds for minimax grid matching with applications to the average case analysis of algorithms , 1989, Comb..

[7]  J. Yukich,et al.  Minimax Grid Matching and Empirical Measures , 1991 .

[8]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[9]  Alison L Gibbs,et al.  On Choosing and Bounding Probability Metrics , 2002, math/0209021.

[10]  C. Villani.  Topics in Optimal Transportation , 2003 .

[11]  C. Villani,et al.  Optimal Transportation and Applications , 2003 .

[12]  C. Villani,et al.  Quantitative Concentration Inequalities for Empirical Measures on Non-compact Spaces , 2005, math/0503123.

[13]  Patrizio Frosini,et al.  Using matching distance in size theory: A survey , 2006, Int. J. Imaging Syst. Technol..

[14]  Ulrike von Luxburg,et al.  A tutorial on spectral clustering , 2007, Stat. Comput..

[15]  Thierry Champion,et al.  The ∞-Wasserstein Distance: Local Solutions and Existence of Optimal Transport Maps , 2008, SIAM J. Math. Anal..

[16]  Emmanuel Boissard.  Simple Bounds for the Convergence of Empirical and Occupation Measures in 1-Wasserstein Distance , 2011, 1103.3188.

[17]  Thibaut Le Gouic,et al.  On the mean speed of convergence of empirical and occupation measures in Wasserstein distance , 2011, 1105.5263.

[18]  A. Guillin,et al.  On the rate of convergence in Wasserstein distance of the empirical measure , 2013, 1312.2128.

[19]  C. Bordenave,et al.  Combinatorial Optimization Over Two Random Point Sets , 2011, 1103.2734.

[20]  Nicolás García Trillos,et al.  On the rate of convergence of empirical measures in $\infty$-transportation distance , 2014, 1407.1157.

[21]  S. Sethuraman,et al.  Consistency of modularity clustering on random geometric graphs , 2016, The Annals of Applied Probability.

[22]  Dejan Slepcev,et al.  A variational approach to the consistency of spectral clustering , 2015, Applied and Computational Harmonic Analysis.