The Entropy Power Inequality and Mrs. Gerber´s Lemma for groups of order 2n

Shannon´s Entropy Power Inequality (EPI) can be viewed as characterizing the minimum differential entropy achievable by the sum of two independent random variables with fixed differential entropies. The EPI is a powerful tool and has been used to resolve a number of problems in information theory. In this paper we examine the existence of a similar entropy inequality for discrete random variables. We obtain an entropy power inequality for random variables taking values in any group of order 2ⁿ, i.e. for such a group G we explicitly characterize the function f_G(x, y) giving the minimum entropy of the group product of two independent G-valued random variables with respective entropies x and y. Random variables achieving the extremum in this inequality are thus the analogs of Gaussians, and these are also determined. It turns out that f_G(x, y) is convex in x for fixed y and, by symmetry, convex in y for fixed x. This is a generalization to groups of order 2ⁿ of the result known as Mrs. Gerber´s Lemma.