Convex cones of generalized multiply monotone functions and the dual cones

TL;DR

This paper characterizes convex cones of generalized multiply monotone functions and their duals, providing insights useful for probability applications, especially in cases where traditional integrability conditions do not hold.

Contribution

It introduces a general framework for describing convex cones of functions with monotone derivatives and characterizes their dual cones, extending previous results to broader settings.

Findings

Characterization of convex cones of functions with monotone derivatives

Description of the dual cones to these convex cones

Applicability to probability theory without restrictive integrability conditions

Abstract

Let $n$ and $k$ be nonnegative integers such that $1\le k\le n+1$. The convex cone $\mathcal{F}_+^{k:n}$ of all functions $f$ on an arbitrary interval $I\subseteq\mathbb{R}$ whose derivatives $f^{(j)}$ of orders $j=k-1,\dots,n$ are nondecreasing is characterized in terms of extreme rays of the cone $\mathcal{F}_+^{k:n}$. A simple description of the convex cone dual to $\mathcal{F}_+^{k:n}$ is given. These results are useful in, and were motivated by, applications in probability. In fact, the results are obtained in a more general setting with certain generalized derivatives of $f$ of the $j$th order in place of $f^{(j)}$. Somewhat similar results were previously obtained in the case when the left endpoint of the interval $I$ is finite, with certain additional integrability conditions; such conditions fail to hold in the mentioned applications.