Convexity of Fisher information with respect to Gaussian perturbation

Let X be an arbitrary continuous random variable and Z be an independent Gaussian random variable with zero mean and unit variance. 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. Following this result, 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 negative otherwise; the second is that log J(X + √tZ) is convex in t.