A new proof of Williamson's representation of multiply monotone functions

This paper provides an alternative proof of the characterization of multiply monotone functions as integrals of simple polynomial-type applications with respect to a probability measure. This constitutes an analogue of the Bernstein-Widder representation of completely monotone functions as Laplace transforms. The proof given here relies on the abstract representation result of Choquet rather than the analytic derivation originally given by Williamson. To this end, we identify the extreme points in the convex set of multiply monotone functions. Our result thus gives a geometric perspective to Williamson's representation.