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.
"A coupling proof of convex ordering for compound distributions." Electron. Commun. Probab. 25 1 - 9, 2020. https://doi.org/10.1214/20-ECP323