On the Continuity of Center-Outward Distribution and Quantile Functions

TL;DR

This paper proves the continuity and invertibility of the center-outward quantile function in multivariate distribution theory, which is essential for defining nested quantile contours in higher dimensions.

Contribution

It establishes the crucial mathematical properties of continuity and invertibility for the center-outward quantile function, advancing the theoretical foundation of multivariate distribution analysis.

Findings

Proves continuity of the center-outward quantile function outside the origin.

Establishes invertibility of the quantile function in the same region.

Supports the existence of nested quantile contours in multivariate distributions.

Abstract

To generalize the notion of distribution function to dimension $d\geq 2$, 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.