Conditions for equality and stability in Shannon's and Tao's entropy power inequalities

Abstract

We show that there is equality in Shannon's Entropy Power Inequality (EPI) if and only if the random variables involved are Gaussian, assuming nothing beyond the existence of differential entropies. This is done by justifying de Bruijn's identity without a second moment assumption. Part of the proof also relies on a re-examination of an example of Bobkov and Chistyakov (2015), which shows that there exists a random variable $X$ with finite differential entropy $h(X),$ such that $h(X+Y) = \infty$ for any independent random variable $Y$ with finite entropy. We prove that either $X$ has this property, or $h(X+Y)$ is finite for any independent $Y$ that does not have this property. Using this, we prove the continuity of $t \mapsto h(X+\sqrt{t}Z)$ at $t=0$, where $Z \sim \mathcal{N}(0,1)$ is independent of $X$, under minimal assumptions. We then establish two stability results: A qualitative stability result for Shannon's EPI in terms of weak convergence under very mild moment conditions, and a quantitative stability result in Tao's discrete analogue of the EPI under log-concavity. The proof for the first stability result is based on a compactness argument, while the proof of the second uses the Cheeger inequality and leverages concentration properties of discrete log-concave distributions.