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A lower bound on the relative entropy with respect to a symmetric probability. (English) Zbl 1320.60056
Summary: 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 $H(\mu|\rho)\geq \frac{\biggl(\int_{\mathbb R}zd\mu(z)\biggr)^2}{2\int_{\mathbb R}z^2d\mu(z)}$ with equality if and only if $$\mu=\rho$$. We give an application to the Curie-Weiss model of self-organized criticality.
##### MSC:
 60E15 Inequalities; stochastic orderings 60F10 Large deviations 94A17 Measures of information, entropy
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