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For the basic case of $L_2$ optimal transport between two probability measures on a Euclidean space, the regularity of the coupling measure and the transport map in the tail regions of these measures is studied.
Convex functions and upper semicontinuous collections
Anderson, R. and V. Klee Jr. (1952) · 1952
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Convex Analysis
Rockafellar, R. T. (1970) · 1970
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On the optimal mapping of distributions
Knott, M. and C. Smith (1984) · 1984
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Regular Variation
Bingham, N., C. Goldie, and J. Teugels (1987) · 1987
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Décomposition polaire et rérrangement monotone des champs de vecteurs
Brenier, Y. (1987) · 1987
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Extreme Values, Regular Variation, and Point Processes
Resnick, S. I. (1987) · 1987
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Polar factorization and monotone rearrangement of vector-valued functions
Brenier, Y. (1991) · 1991
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Existence and uniqueness of monotone measure-preserving maps
McCann, R. J. (1995) · 1995
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Optimal transportation plans and convergence in distribution
Cuesta-Albertos, J., C. Matrán, and A. Tuero-Diaz (1997) · 1997
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Alberti, G. and L. Ambrosio (1999) · 1999
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Graphical convergence of sums of monotone mappings
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Topics in Optimal Transportation
Villani, C. (2003) · 2003
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Extremal behavior of regularly varying stochastic processes
Hult, H. and F. Lindskog (2005) · 2005
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Regular variation for measures on metric spaces
Hult, H. and F. Lindskog (2006) · 2006
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Heavy-Tail Phenomena. Probabilistic and Statistical Modeling
Resnick, S. I. (2007) · 2007
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Optimal Transport. Old and New
Villani, C. (2009) · 2009
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Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps
Lindskog, F., S. I. Resnick, and J. Roy (2014) · 2014
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Monge–Kantorovich depth, quantiles, ranks and signs
Chernozhukov, V., A. Galichon, M. Hallin, and M. Henry (2017) · 2017
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Center-outward distribution functions, quantiles, ranks, and signs in
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Molchanov, I. (2005) · 2005
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Extreme Value Theory. An Introduction
de Haan, L. and A. Ferreira (2006) · 2006
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del Barrio, E., J. A. Cuesta-Albertos, M. Hallin, and C. Matrán (2018) · 2018
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Central limit theorems for empirical transportation cost in general dimension
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