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 e √ tZ) is concave in t, where the proof hinged on the first and second order derivatives of h(X + √ tZ). Specifically, these two derivatives are signed, i.e., ∂ ∂t h(X + √ tZ) ≥ 0 and ∂ 2 ∂t2 h(X + √ tZ) ≤ 0. In this paper, we show that the third order derivative of h(X + √ tZ) is nonnegative, which implies that the Fisher information J(X + √ tZ) is convex in t. We further show that the fourth order derivative of h(X + √ tZ) is nonpositive. Following these results, we make two conjectures on h(X+ √ tZ): the first is that ∂ n ∂tn h(X+ √ tZ) is nonnegative in t if n is odd, and nonpositive otherwise; the second is that log J(X + √ tZ) is convex in t. The concavity of h( √ tX + √ 1− tZ) is studied, revealing its connection with Costa’s EPI.