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This work is devoted to the study of rates of convergence of the empirical measures μn over a sample (Xk) of independent identically distributed real-valued random variables towards the common distribution μ in Kantorovich transport distances Wp. The focus is on finite range bounds on the expected Kantorovich distances E(Wp(μn, μ)) in terms of moments and analytic conditions on the measure μ and its distribution function. The study describes a variety of rates, from the standard one to slower rates, and both lower and upper-bounds on E(Wp(μn,μ)) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, log-concavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.
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Equal area partitions of the sphere with diameter bounds, via optimal transport
Abstract We prove existence of equal area partitions of the unit sphere via optimal transport methods, accompanied by diameter bounds written in terms of Monge–Kantorovich distances. This can be used to obtain bounds on the expectation of the maximum diameter of partition sets, when points are uniformly sampled from the sphere. An application to the computation of sliced Monge–Kantorovich distances is also presented.
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- Award ID(s):
- 2000128
- PAR ID:
- 10576357
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Bulletin of the London Mathematical Society
- Volume:
- 57
- Issue:
- 5
- ISSN:
- 0024-6093
- Format(s):
- Medium: X Size: p. 1524-1538
- Size(s):
- p. 1524-1538
- Sponsoring Org:
- National Science Foundation
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