Sharp variance-entropy comparison for nonnegative Gaussian quadratic forms

TL;DR

This paper demonstrates that among certain sums and quadratic forms of gamma and Gaussian variables, the configurations with equal weights maximize differential entropy under fixed variance, providing sharp bounds and implications for information theory.

Contribution

It establishes that equal coefficient quadratic forms maximize entropy among nonnegative forms, offering sharp bounds for relative entropy and channel capacity estimates.

Findings

Equal coefficient sums maximize differential entropy for gamma variables.

Diagonal quadratic forms with equal weights maximize entropy among nonnegative Gaussian forms.

Provides sharp lower bounds for relative entropy and channel capacity under gamma noise.

Abstract

In this article we study weighted sums of $n$ i.i.d. Gamma($\alpha$) random variables with nonnegative weights. We show that for $n \geq 1/\alpha$ the sum with equal coefficients maximizes differential entropy when variance is fixed. As a consequence, we prove that among nonnegative quadratic forms in $n$ independent standard Gaussian random variables, a diagonal form with equal coefficients maximizes differential entropy, under a fixed variance. This provides a sharp lower bound for the relative entropy between a nonnegative quadratic form and a Gaussian random variable. Bounds on capacities of transmission channels subject to $n$ independent additive gamma noises are also derived.