Bernoulli sums and Rényi entropy inequalities

Abstract

We investigate the Rényi entropy of sums of independent integer-valued random variables through Fourier theoretic means, and give sharp comparisons between the variance and the Rényi entropy for sums of independent Bernoulli random variables. As applications, we prove that a discrete “min-entropy power” is superadditive with respect to convolution modulo a universal constant, and give new bounds on an entropic generalization of the Littlewood-Offord problem that are sharp in the “Poisson regime”.