Higher-order Monotone Functions and Probability Theory

Abstract

From topological properties of the cone of real functions with higher order monotonicity properties, we derive an extreme point Choquet representation theorem; this, in fact, follows from Krein-Milman’s theory, since there is a natural compact convex base. This fills the gap between convexity and complete monotonicity, in the sense that we obtain integral representations for functions such that (-l) k f (k) (x) ≥ 0, k = 0,1,…, n (in the form of special “beta transforms”), that in the case of completely monotone functions is Bernstein’s classical result about Laplace transforms of non-decreasing functions. The integral representation is used to reobtain Khinchine’s characterization of unimodal distribution functions and Polya’s characteristic functions. On the other hand, we obtain extended classes of unimodal distributions (different from the star-shaped unimodality of Olshen and Savage), and put in a proper light Mejzler’s classes in extreme value theory, and special classes of characteristic functions such as Sakovic’s class. Using fractional calculus, we have straightforward extensions of the previous results, and consider generalized beta transforms, naturally associated with order statistics with general parent distribution.