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# On the Rate of Convergence of Empirical Measures in $\infty$-transportation Distance

Published:2015-03-02
Printed: Dec 2015
• Nicolas Garcia Trillos,
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA.
• Dejan Slepcev,
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA.
 Format: LaTeX MathJax PDF

## Abstract

We consider random i.i.d. samples of absolutely continuous measures on bounded connected domains. We prove an upper bound on the $\infty$-transportation distance between the measure and the empirical measure of the sample. The bound is optimal in terms of scaling with the number of sample points.
 Keywords: rate, convergence
 MSC Classifications: 01B01 - unknown classification 01B01

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