An extremal property of the normal distribution, with a discrete analog

TL;DR

This paper characterizes the normal and Poisson distributions as extremal cases of strong log-concavity, using inequalities like Brascamp-Lieb and Chebyshev's, linking these properties to approximation methods.

Contribution

It provides a novel characterization of Gaussian and Poisson measures through strong log-concavity and inequalities, connecting these to approximation techniques.

Findings

Gaussian measure uniquely strongly log-concave with its covariance

Poisson measure characterized via Chebyshev's inequality

Results relate to Stein and Bakry-Emery methods

Abstract

We prove, using the Brascamp-Lieb inequality, that the Gaussian measure is the only strong log-concave measure having a strong log-concavity parameter equal to its covariance matrix. We also give a similar characterization of the Poisson measure in the discrete case, using "Chebyshev's other inequality". We briefly discuss how these results relate to Stein and Stein-Chen methods for Gaussian and Poisson approximation, and to the Bakry-Emery calculus.