# An Entropy Power Inequality for Discrete Random Variables

## TL;DR

This paper establishes a new entropy power inequality for discrete random variables, showing that the sum of their entropy powers is bounded by twice the entropy power of their sum, with a proof leveraging perturbation and continuous inequalities.

## Contribution

The paper introduces the first entropy power inequality for discrete variables, extending concepts from continuous entropy power inequalities to the discrete setting.

## Key findings

- The inequality is tight for certain distributions.
- The proof uses perturbation with continuous variables.
- Provides a new tool for analyzing discrete entropy.

## Abstract

Let $\mathsf{N}_{\rm d}\left[X\right]=\frac{1}{2\pi {\rm e}}{\rm e}^{2\mathsf{H}\left[X\right]}$ denote the entropy power of the discrete random variable $X$ where $\mathsf{H}\left[X\right]$ denotes the discrete entropy of $X$. In this paper, we show that for two independent discrete random variables $X$ and $Y$, the entropy power inequality $\mathsf{N}_{\rm d}\left[X\right]+\mathsf{N}_{\rm d}\left[Y\right]\leq 2 \mathsf{N}_{\rm d}\left[X+Y\right]$ holds and it can be tight. The basic idea behind the proof is to perturb the discrete random variables using suitably designed continuous random variables. Then, the continuous entropy power inequality is applied to the sum of the perturbed random variables and the resulting lower bound is optimized.