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On concentration of the empirical measure for radial transport costs

Author

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  • Larsson, Martin
  • Park, Jonghwa
  • Wiesel, Johannes

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.

Suggested Citation

  • Larsson, Martin & Park, Jonghwa & Wiesel, Johannes, 2024. "On concentration of the empirical measure for radial transport costs," Stochastic Processes and their Applications, Elsevier, vol. 178(C).
    • Handle: RePEc:eee:spapps:v:178:y:2024:i:c:s0304414924001728
    DOI: 10.1016/j.spa.2024.104466
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    References listed on IDEAS

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    1. Dedecker, Jérôme & Fan, Xiequan, 2015. "Deviation inequalities for separately Lipschitz functionals of iterated random functions," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 60-90.
    2. Bartlett, Peter & Lugosi, Gábor, 1999. "An inequality for uniform deviations of sample averages from their means," Statistics & Probability Letters, Elsevier, vol. 44(1), pages 55-62, August.
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