Entropy power inequality for a family of discrete random variables

TL;DR

This paper extends the entropy power inequality (EPI) to certain discrete distributions, especially binomial distributions, and shows it holds for sums of many IID discrete variables, broadening its applicability.

Contribution

It identifies conditions under which the discrete EPI holds for binomial distributions and sums of IID discrete variables, expanding previous results.

Findings

EPI holds for binomial pairs (B(m,p), B(n,p)) with large m,n for p=0.5

EPI applies to sums of many IID discrete variables

Established n_0(p) such that EPI holds for all m,n ≥ n_0(p)

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\"{e}s and Vignat showed that it holds for the pair (B(m,p), B(n,p)), m,n \in \mathbb{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 n_0(p) such that for all m,n \geq n_0(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.