Log-concavity and discrete degrees of freedom

TL;DR

This paper introduces the concept of discrete degrees of freedom for log-concave sequences and demonstrates extremal properties of geometric and Poisson distributions regarding entropy and probability at the mean.

Contribution

It defines discrete degrees of freedom for log-concave sequences and proves geometric distribution minimizes Rènyi entropy, while Poisson maximizes the probability at the mean for ultra-log-concave variables.

Findings

Geometric distribution minimizes Rènyi entropy of order infinity among discrete log-concave variables.

Poisson distribution maximizes the probability at the mean among ultra-log-concave variables.

Introduces the notion of discrete degrees of freedom for log-concave sequences.

Abstract

We develop the notion of discrete degrees of freedom of a log-concave sequence and use it to prove that geometric distribution minimises R\'enyi entropy of order infinity under fixed variance, among all discrete log-concave random variables in $\mathbb{Z}$. We also show that the quantity $\mathbb{P}(X=\mathbb{E} X)$ is maximised, among all ultra-log-concave random variables with fixed integral mean, for a Poisson distribution.