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On the Rate of Convergence of Empirical Measures in ∞-transportation Distance

Published online by Cambridge University Press:  20 November 2018

Nicolás García Trillos  and
Dejan Slepčev
Nicolás García Trillos
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA. e-mail: ngarciat@andrew.cmu.edu, slepcev@math.cmu.edu
Dejan Slepčev
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA. e-mail: ngarciat@andrew.cmu.edu, slepcev@math.cmu.edu
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Abstract

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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

60B1060D0505C70optimal transportationoptimal matchinginfinity transportation distancemin-max distanceempirical measure
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 6 , 01 December 2015 , pp. 1358 - 1383
Copyright
Copyright © Canadian Mathematical Society 2015

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