A concavity property for the reciprocal of Fisher information and its consequences on Costa's EPI

Abstract: We prove that the reciprocal of Fisher information of a log-concave probability density $X$ in ${\bf{R}}<sup>n$ is concave in $t$ with respect to the addition of a Gaussian noise $Z_t = N(0, tI_n)$. As a byproduct of this result we show that the third derivative of the entropy power of a log-concave probability density $X$ in ${\bf{R}}<sup>n$ is nonnegative in $t$ with respect to the addition of a Gaussian noise $Z_t$. For log-concave densities this improves the well-known Costa's concavity property of the entropy power.