Sharp Variance-Entropy Comparison for Nonnegative Gaussian Quadratic Forms

In this article we study weighted sums of n i.i.d. Gamma(alpha) random variables with nonnegative weights. We show that for n >= 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 subjects to n independent additive gamma noises are also derived.