Graphical and uniform consistency of estimated optimal transport plans

Johan Segers ()
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Johan Segers: Université catholique de Louvain, LIDAM/ISBA, Belgium

No 2022022, LIDAM Discussion Papers ISBA from Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA)

Abstract: A general theory is provided delivering convergence of maximal cyclically monotone mappings containing the supports of coupling measures of sequences of pairs of possibly random probability measures on Euclidean space. The theory is based on the identification of such a mapping with a closed subset of a Cartesian product of Euclidean spaces and leveraging tools from random set theory. Weak convergence in the appropriate Fell space together with the maximal cyclical monotonicity then automatically yields local uniform convergence of the associated mappings. Viewing such mappings as optimal transport plans between probability measures with respect to the squared Euclidean distance as cost function yields consistency results for notions of multivariate ranks and quantiles based on optimal transport, notably the empirical center-outward distribution and quantile functions.

Keywords: Convex function; coupling measure; distribution function; Fell topology; maximal monotone mapping; quantile function (search for similar items in EconPapers)
Pages: 27
Date: 2022-08-04
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Citations: View citations in EconPapers (1)

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