The stochastic order of probability measures on ordered metric spaces

TL;DR

This paper introduces a stochastic order on probability measures over ordered metric spaces, establishing its properties and applications, including inequalities for positive matrices, advancing the mathematical understanding of probabilistic orderings.

Contribution

It defines and analyzes a stochastic order on probability measures in ordered metric spaces, proving its antisymmetry, closedness, and exploring order-completeness and inequalities.

Findings

The stochastic order is a closed partial order on probability measures.

Antisymmetry and closedness are established for measures on Banach spaces with normal cones.

Derived inequalities include arithmetic-geometric-harmonic mean inequalities for positive matrices.

Abstract

The general notion of a stochastic ordering is that one probability distribution is smaller than a second one if the second attaches more probability to higher values than the first. Motivated by recent work on barycentric maps on spaces of probability measures on ordered Banach spaces, we introduce and study a stochastic order on the space of probability measures $\mathcal{P}(X)$, where $X$ is a metric space equipped with a closed partial order, and derive several useful equivalent versions of the definition. We establish the antisymmetry and closedness of the stochastic order (and hence that it is a closed partial order) for the case of a partial order on a Banach space induced by a closed normal cone with interior. We also consider order-completeness of the stochastic order for a cone of a finite-dimensional Banach space and derive a version of the arithmetic-geometric-harmonic mean inequalities in the setting of the associated probability space on positive matrices.