A universal, canonical dispersive ordering in metric spaces

Abstract

The main goal of this paper is to study a new dispersive order in an arbitrary metric space. More precisely, given two random variables on a metric space, the order decides which one has a more concentrated distribution. It is motivated by the wish to be able to make new comparisons that the currently defined orderings cannot make.Related to this problem, several statistical parameters of a random variable, giving different kinds of information about the distribution, are studied. It is proved that the new dispersive order also compares some of these parameters. Special attention is paid to Riemannian manifolds, a natural class of metric spaces. Here, the new order behaves in a distinguished way: the differentiability axioms allow obtaining several equations which facilitate the decision.The principal properties of the order are presented, as well as several applications. Finally, the independence of this order from those already known is proved.