A Coupling Proof of Convex Ordering for Compound Distributions

2019/10/16 by Bérard, Jean, Juillet, Nicolas
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1910.07218

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.

Related

• Does the Convex Order Between the Distributions of Linear Functionals Imply the Convex Order Between the Probability Distributions Over \mathbb R^d?
• Convex comparison of Gaussian mixtures
• On convex order and supermodular order without finite mean
• Convex transform order of Beta distributions with some consequences
• Sharp convex bounds on the aggregate sums--An alternative proof