Convex comparison of Gaussian mixtures

TL;DR

This paper investigates the convex order comparison between Gaussian distributions and Gaussian mixtures, providing necessary and sufficient conditions for such comparisons based on stochastic differential equations.

Contribution

It introduces intrinsic conditions for convex ordering between Gaussian and Gaussian mixture distributions, advancing understanding of distribution comparison in stochastic processes.

Findings

Derived necessary and sufficient conditions for convex order comparison.

Found close relationships between the conditions in studied examples.

Enhanced theoretical understanding of distribution propagation in stochastic differential equations.

Abstract

Motivated by the study of the propagation of convexity by semi-groups of stochastic differential equations and convex comparison between the distributions of solutions of two such equations, we study the comparison for the convex order between a Gaussian distribution and a Gaussian mixture. We give and discuss intrinsic necessary and sufficient conditions for convex ordering. On the examples that we have worked out, the two conditions appear to be closely related.