Beyond the entropy power inequality, via rearrangements

TL;DR

This paper introduces a new lower bound on the Rényi differential entropy of sums of independent random vectors using rearrangement techniques, improving upon classical bounds and offering new proofs and applications.

Contribution

It presents a novel lower bound based on rearrangements that surpasses the entropy power inequality for Shannon entropy and provides new proofs and applications.

Findings

Lower bound on Rényi entropy via rearrangements

Improved bounds over classical entropy power inequality

Applications to symmetrization of Lévy processes

Abstract

A lower bound on the R\'enyi differential entropy of a sum of independent random vectors is demonstrated in terms of rearrangements. For the special case of Boltzmann-Shannon entropy, this lower bound is better than that given by the entropy power inequality. Several applications are discussed, including a new proof of the classical entropy power inequality and an entropy inequality involving symmetrization of L\'evy processes.