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

• We consider the entropy power of the sum X+ZtX+Zt, where ZtZt is a Gaussian noise.
• If XX has a log-concave density. We extend Costa’s result on concavity.
• For log-concave densities the third derivative of entropy power has a positive sign.
• Also, the reciprocal of Fisher information of X+ZtX+Zt is concave.

We prove that the reciprocal of Fisher information of a log-concave probability density XX in RnRn is concave in tt with respect to the addition of a Gaussian noise Zt=N(0,tIn)Zt=N(0,tIn). As a byproduct of this result we show that the third derivative of the entropy power of a log-concave probability density XX in RnRn is nonnegative in tt with respect to the addition of a Gaussian noise ZtZt. For log-concave densities this improves the well-known Costa’s concavity property of the entropy power (Costa, 1985).