A lower bound on the relative entropy with respect to a symmetric probability
Résumé
Let ρ and µ be two probability measures on R which are not the Dirac mass at 0. We denote by H(µ|ρ) the relative entropy of µ with respect to ρ. We prove that, if ρ is symmetric and µ has a finite first moment, then H(µ|ρ) ≥ R z dµ(z) 2 2 R z 2 dµ(z) , with equality if and only if µ = ρ. We give an applicaion to the Curie-Weiss model of self-organized criticality.
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