2026-05-05T13:04:56Zhttps://uvadoc.uva.es/oai/request
oai:uvadoc.uva.es:10324/823972026-03-11T07:42:58Zcom_10324_1151com_10324_931com_10324_894col_10324_1278
UVaDOC author Barrio Tellado, Eustasio del author González Sanz, Alberto author Loubes, Jean-Michel 2026-01-30T10:40:15Z 2026-01-30T10:40:15Z 2024 Bernoulli 30 (1) 554 - 580, February 2024 1350-7265 https://uvadoc.uva.es/handle/10324/82397 10.3150/23-BEJ1608 554 1 580 Bernoulli 30 We prove a Central Limit Theorem for the empirical optimal transport cost, √nmn+m{Tc(Pn,Qm)−Tc(P,Q)}, in the semi-discrete case, i.e when the distribution P is supported in N points, but without assumptions on Q. We show that the asymptotic distribution is the sup of a centered Gaussian process, which is Gaussian under some additional conditions on the probability Q and on the cost. Such results imply the central limit theorem for the p-Wassertein distance, for p≥1. This means that, for fixed N, the curse of dimensionality is avoided. To better understand the influence of such N, we provide bounds of E|Wpp(P,Qm)−Wpp(P,Q)| depending on m and N. Finally, the semi-discrete framework provides a control on the second derivative of the dual formulation, which yields the first central limit theorem for the optimal transport potentials and Laguerre cells. The results are supported by simulations that help to visualize the given limits and bounds. We analyse also the cases where classical bootstrap works. spa Estadística Probabilidad Central limit theorems for semi-discrete Wasserstein distances info:eu-repo/semantics/article 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 URL https://uvadoc.uva.es/bitstream/10324/82397/1/2202.06380v1.pdf File MD5 575a5410e652207d759a1a376e324c1f 1417601 application/pdf 2202.06380v1.pdf