Entropy, subentropy and the elementary symmetric functions

TL;DR

This paper investigates the mathematical properties of entropy and subentropy functions expressed through elementary symmetric polynomials, revealing their derivatives' relationships and their connection to infinitely divisible distributions.

Contribution

It introduces complex contour integral techniques to analyze entropy and subentropy, uncovering new properties and their links to probability distributions.

Findings

Derivatives of -Q equal derivatives of H of one higher order.

First derivatives of H and Q are completely monotone functions.

exp(-H) and exp(-Q) are Laplace transforms of infinitely divisible distributions.

Abstract

We use complex contour integral techniques to study the entropy H and subentropy Q as functions of the elementary symmetric polynomials, revealing a series of striking properties. In particular for these variables, derivatives of -Q are equal to derivatives of H of one higher order and the first derivatives of H and Q are seen to be completely monotone functions. It then follows that exp (-H) and exp(-Q) are Laplace transforms of infinitely divisible probability distributions.