A lower bound on the relative entropy with respect to a symmetric probability

Abstract

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 (Formula presented) with equality if and only if μ=ρ. We give an applicaion to the Curie-Weiss model of self-organized criticality.