A Coupling Proof of Convex Ordering for Compound Distributions

TL;DR

This paper presents a new coupling proof demonstrating that convex ordering of summand distributions implies convex ordering of their compound distributions, providing a concrete probabilistic coupling approach.

Contribution

It offers an alternative, coupling-based proof of a classical convex ordering result in risk theory, using martingale law representations.

Findings

Convex ordering of summands implies convex ordering of compounds.

Provides a concrete coupling construction for the proof.

Utilizes mixture representations of discrete martingale laws.

Abstract

In this paper, we give an alternative proof of the fact that, when compounding a nonnegative probability distribution, convex ordering between the distributions of the number of summands implies convex ordering between the resulting compound distributions. Although this is a classical textbook result in risk theory, our proof exhibits a concrete coupling between the compound distributions being compared, using the representation of one-period discrete martingale laws as a mixture of the corresponding extremal measures.