On the problem of reversibility of the entropy power inequality

TL;DR

This paper investigates the limits of reversing the entropy power inequality, showing it is impossible for all convex distributions and discussing related phenomena for identical distributions.

Contribution

It proves that the reversibility of the entropy power inequality cannot hold universally for convex probability distributions.

Findings

Reversibility is impossible for the entire class of convex distributions.

Reversibility can be achieved under certain conditions for specific distributions.

Related phenomena are discussed for identically distributed summands.

Abstract

As was shown recently by the authors, the entropy power inequality can be reversed for independent summands with sufficiently concave densities, when the distributions of the summands are put in a special position. In this note it is proved that reversibility is impossible over the whole class of convex probability distributions. Related phenomena for identically distributed summands are also discussed.