Lower Bounds on Multivariate Higher Order Derivatives of Differential Entropy.

This paper studies the properties of the derivatives of differential entropy H(Xt) in Costa's entropy power inequality. For real-valued random variables, Cheng and Geng conjectured that for m≥1, (-1)m+1(dm/dtm)H(Xt)≥0, while McKean conjectured a stronger statement, whereby (-1)m+1(dm/dtm)H(Xt)≥(-1)m+1(dm/dtm)H(XGt). Here, we study the higher dimensional analogues of these conjectures.

A Generalization of the Concavity of Rényi Entropy Power.

Recently, Savaré-Toscani proved that the Rényi entropy power of general probability densities solving the -nonlinear heat equation in Rn is a concave function of time under certain conditions of three parameters n,p,μ, which extends Costa's concavity inequality for Shannon's entropy power to the Rényi entropy power. In this paper, we give a condition Φ(n,p,μ) of n,p,μ under which the concavity of the Rényi entropy power is valid. The condition Φ(n,p,μ) contains Savaré-Toscani's condition as a special case and much more cases.