On the Continuity of Center-Outward Distribution and Quantile Functions

05/13/2018

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by   Alessio Figalli, et al.

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To generalize the notion of distribution function to dimension d≥ 2, in the recent papers it was proposed a concept of center-outward distribution function based on optimal transportation ideas, and the inferential properties of the corresponding center-outward quantile function were studied. A crucial tool needed to derive the desired inferential properties is the continuity and invertibility for the center-outward quantile function outside the origin, as this ensures the existence of closed and nested quantile contours. The aim of this paper is to prove such a continuity and invertibility result.