Higher Order Derivatives in Costa's Entropy Power Inequality.

Let X be an arbitrary continuous random variable and Z be an independent Gaussian random variable with zero mean and unit variance. For t > 0, Costa proved that e2h(X+√t Z) is concave in t, where the proof hinged on the first and second order derivatives of h(X + √t Z). In particular, these two derivatives are signed, i.e., (∂/∂t)h(X + √tZ) ≥ 0 and (∂2/∂t2)h(X + √tZ) ≤ 0. In this paper, we show th...