Convex Order Beyond Dimension One: Projection Tests, Counterexamples and Gaussian Mixtures

2026-10-05

AI Summary

This study addresses the insufficiency of projection-based tests for convex order in high dimensions, particularly the theoretical challenge of comparing Gaussian and mixture distributions. By leveraging canonical functions, Popoviciu’s inequality, and multidimensional convex order theory, we rigorously analyze counterexamples involving symmetric distributions and prove the failure of projections in radial settings, while establishing their sufficiency under specific families of linear transformations and two-component Gaussian mixtures. The primary contribution is the first derivation of necessary and sufficient conditions for a finite-dimensional centered Gaussian distribution to be dominated by a two-component Gaussian mixture, thereby filling a theoretical gap left by Jourdain and Pagès. Furthermore, this work confirms that one-dimensional projected convex order is equivalent to multidimensional convex order within this class of models, providing precise criteria for risk quantification.

Abstract

In actuarial science and quantitative finance, convex order provides a natural way to compare risks with the same mean. In dimension one, convex order is well understood through several characterisations. In higher dimensions, a natural approach is to compare all one-dimensional projections, but, although necessary, the resulting condition is in general not sufficient for multivariate convex order. A simple counterexample due to Pinelis exploits Popoviciu's inequality. We revisit this example using gauge functions, and show that gauge functions themselves do not suffice to characterise multivariate convex order for symmetric distributions, even in dimension two. We then show that the radial case makes the source of the failure transparent and construct radial counterexamples in every dimension at least two. Nevertheless, projection tests are sufficient for some important families of distributions. In particular, this is the case for linear transformations of a common radially distributed random vector. This framework includes elliptical distributions generated from a common radial distribution and, in particular, centred Gaussian distributions. Beyond comparisons between Gaussian distributions, Jourdain and Pagès recently studied the comparison of a Gaussian distribution with a Gaussian mixture, obtaining necessary conditions and sufficient conditions whose equivalence was left open in general. In this paper, we settle the comparison of a centred Gaussian distribution with a mixture of two centred Gaussian distributions: in every finite dimension, the Gaussian is dominated by the mixture in multivariate convex order if and only if all one-dimensional projections are ordered in convex order.

Problem

• multivariate convex order
• projection tests
• Gaussian mixtures
• elliptical distributions
• radial distributions

Innovation

• Multivariate Convex Order
• Projection Tests
• Gaussian Mixtures
• Gauge Functions
• Radial Distributions