R\'enyi entropy and variance comparison for symmetric log-concave random variables

TL;DR

This paper characterizes the distributions that minimize Rényi entropy among symmetric log-concave variables with fixed variance, identifying uniform and two-sided exponential distributions depending on the order, and extends results to one-sided exponential distributions.

Contribution

It establishes the minimal Rényi entropy distributions for symmetric log-concave variables across different orders and identifies the exponential distribution as the minimizer for certain cases.

Findings

Uniform distribution minimizes for α in (0, α*]

Two-sided exponential minimizes for α in [α*, ∞)

One-sided exponential minimizes for α ≥ 2 among log-concave variables

Abstract

We show that for any $\alpha>0$ the R\'enyi entropy of order $\alpha$ is minimized, among all symmetric log-concave random variables with fixed variance, either for a uniform distribution or for a two sided exponential distribution. The first case occurs for $\alpha \in (0,\alpha^*]$ and the second case for $\alpha \in [\alpha^*,\infty)$, where $\alpha^*$ satisfies the equation $\frac{1}{\alpha^*-1}\log \alpha^*= \frac12 \log 6$, that is $\alpha^* \approx 1.241$. Using those results, we prove that one-sided exponential distribution minimizes R\'enyi entropy of order $\alpha \geq 2$ among all log-concave random variables with fixed variance.