Entropy power inequality for a family of discrete random variables
Abstract
It is known that the Entropy Power Inequality (EPI) always holds if the random variables have density. Not much work has been done to identify discrete distributions for which the inequality holds with the differential entropy replaced by the discrete entropy. Harremo\"es and Vignat showed that it holds for the pair (B(m,p), B(n,p)), m,n ∈ N, (where B(n,p) is a Binomial distribution with n trials each with success probability p) for p = 0.5. In this paper, we considerably expand the set of Binomial distributions for which the inequality holds and, in particular, identify n0(p) such that for all m,n ≥ n0(p), the EPI holds for (B(m,p), B(n,p)). We further show that the EPI holds for the discrete random variables that can be expressed as the sum of n independent identical distributed (IID) discrete random variables for large n.