  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.

Probability