Generalized Entropy Power Inequalities and Monotonicity Properties of Information

TL;DR

This paper introduces new generalized entropy power and Fisher information inequalities for sums of independent random variables, providing insights into information behavior in the central limit theorem.

Contribution

It presents novel inequalities that relate the information of sums over subsets of variables, extending classical results and simplifying proofs of monotonicity in the CLT.

Findings

New entropy power inequalities for sums of independent variables.

A simple proof of the monotonicity of information in the CLT.

Generalization to sums of independent, not necessarily identical, variables.

Abstract

New families of Fisher information and entropy power inequalities for sums of independent random variables are presented. These inequalities relate the information in the sum of $n$ independent random variables to the information contained in sums over subsets of the random variables, for an arbitrary collection of subsets. As a consequence, a simple proof of the monotonicity of information in central limit theorems is obtained, both in the setting of i.i.d. summands as well as in the more general setting of independent summands with variance-standardized sums.