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2015 A lower bound on the relative entropy with respect to a symmetric probability
Raphaël Cerf, Matthias Gorny
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Electron. Commun. Probab. 20: 1-5 (2015). DOI: 10.1214/ECP.v20-3920

Abstract

Let $\rho$ and $\mu$ be two probability measures on $\mathbb{R}$ which are not the Dirac mass at $0$. We denote by $H(\mu|\rho)$ the relative entropy of $\mu$ with respect to $\rho$. We prove that, if $\rho$ is symmetric and $\mu$ has a finite first moment, then \[ H(\mu|\rho)\geq \frac{\displaystyle{\left(\int_{\mathbb{R}}z\,d\mu(z)\right)^2}}{\displaystyle{2\int_{\mathbb{R}}z^2\,d\mu(z)}}\,,\] with equality if and only if $\mu=\rho$. We give an applicaion to the Curie-Weiss model of self-organized criticality.

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Raphaël Cerf. Matthias Gorny. "A lower bound on the relative entropy with respect to a symmetric probability." Electron. Commun. Probab. 20 1 - 5, 2015. https://doi.org/10.1214/ECP.v20-3920

Information

Accepted: 13 January 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 1320.60056
MathSciNet: MR3304411
Digital Object Identifier: 10.1214/ECP.v20-3920

Subjects:
Primary: 60E15
Secondary: 60F10, 94A17

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