Stability and tail limits of transport-based quantile contours
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We extend Robert McCann's treatment of the existence and uniqueness of an optimal transport map between two probability measures on a Euclidean space to a class of possibly infinite measures, finite outside neighbourhoods of the origin. For convergent sequences of pairs of such measures, we study the stability of the multivalued transport maps and associated quantile contours, defined as the images of spheres under these maps. The measures involved in the coupling are not required to be absolutely continuous, there are no restrictions on their supports, and no moment assumptions are needed. Weakly convergent sequences of probability measures forming a special case, our results apply to the recently introduced Monge--Kantorovich depth contours. The set-up involving infinite limit measures is applied to regularly varying probability measures: we derive tail limits of transport maps and of quantile contours defined with respect to a judiciously chosen spherical reference measure. Examples are discussed in detail.
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