{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,27]],"date-time":"2026-09-27T09:19:23Z","timestamp":1790500763380,"version":"4.1.0"},"reference-count":57,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2021,2,15]],"date-time":"2021-02-15T00:00:00Z","timestamp":1613347200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2022,4]]},"abstract":"<jats:p>The theory of optimal transportation has experienced a sharp increase in interest in many areas of economic research such as optimal matching theory and econometric identification. A particularly valuable tool, due to its convenient representation as the gradient of a convex function, has been the Brenier map: the matching obtained as the optimizer of the Monge\u2013Kantorovich optimal transportation problem with the euclidean distance as the cost function. Despite its popularity, the statistical properties of the Brenier map have yet to be fully established, which impedes its practical use for estimation and inference. This article takes a first step in this direction by deriving a convergence rate for the simple plug-in estimator of the potential of the Brenier map via the semi-dual Monge\u2013Kantorovich problem. Relying on classical results for the convergence of smoothed empirical processes, it is shown that this plug-in estimator converges in standard deviation to its population counterpart under the minimax rate of convergence of kernel density estimators if one of the probability measures satisfies the Poincar\u00e9 inequality. Under a normalization of the potential, the result extends to convergence in the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466621000037_inline1.png\"\/><jats:tex-math>\n$L^2$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> norm, while the Poincar\u00e9 inequality is automatically satisfied. The main mathematical contribution of this article is an analysis of the second variation of the semi-dual Monge\u2013Kantorovich problem, which is of independent interest.<\/jats:p>","DOI":"10.1017\/s0266466621000037","type":"journal-article","created":{"date-parts":[[2021,2,15]],"date-time":"2021-02-15T10:04:39Z","timestamp":1613383479000},"page":"381-417","source":"Crossref","is-referenced-by-count":5,"title":["ON THE CONVERGENCE RATE OF POTENTIALS OF BRENIER MAPS"],"prefix":"10.1017","volume":"38","author":[{"given":"Florian F.","family":"Gunsilius","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2021,2,15]]},"reference":[{"key":"S0266466621000037_r45","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-20828-2"},{"key":"S0266466621000037_r4","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0837797-X"},{"key":"S0266466621000037_r55","volume-title":"Optimal Transport: Old and New","author":"Villani","year":"2008"},{"key":"S0266466621000037_r41","first-page":"41","article-title":"Higher derivatives of conjugate convex functions","volume":"1","author":"Rockafellar","year":"1977","journal-title":"Journal of Applied Analysis"},{"key":"S0266466621000037_r27","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-007-0087-9"},{"key":"S0266466621000037_r30","doi-asserted-by":"crossref","unstructured":"Gunsilius, F.F. & Schennach, S.M. 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