On concentration of the empirical measure for radial transport costs

Martin Larsson, Jonghwa Park and Johannes Wiesel

Stochastic Processes and their Applications, 2024, vol. 178, issue C

Abstract: Let μ be a probability measure on Rd and μN its empirical measure with sample size N. We prove a concentration inequality for the optimal transport cost between μ and μN for radial cost functions with polynomial local growth, that can have superpolynomial global growth. This result generalizes and improves upon estimates of Fournier and Guillin. The proof combines ideas from empirical process theory with known concentration rates for compactly supported μ. By partitioning Rd into annuli, we infer a global estimate from local estimates on the annuli and conclude that the global estimate can be expressed as a sum of the local estimate and a mean-deviation probability for which efficient bounds are known.

Keywords: Empirical measures; Wasserstein distances; Optimal transport cost; Empirical process theory; Concentration inequalities; Polynomial local growth (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1016/j.spa.2024.104466

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