2014/09/19 by Fan Cheng, Cheng, Fan, Yanlin Geng +1
Decision Sciences · Mathematics · Physics and Astronomy · #FOS: Computer and information sciences #Information Theory (cs.IT) #Mathematical Inequalities and Applications #Probabilistic and Robust Engineering Design #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.1409.5543
openalex publication_date 2014/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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). Specifically, these two derivatives are signed, i.e., (∂)/(∂ t)h(X+√(t)Z) ≥ 0 and (∂2)/(∂ t2)h(X+√(t)Z) ≤ 0. In this paper, we show that the third order derivative of h(X+√(t)Z) is nonnegative, which implies that the Fisher information J(X+√(t)Z) is convex in t. We further show that the fourth order derivative of h(X+√(t)Z) is nonpositive. Following the first four derivatives, we make two conjectures on h(X+√(t)Z): the first is that (∂n)/(∂ tn) h(X+√(t)Z) is nonnegative in t if n is odd, and nonpositive otherwise; the second is that log J(X+√(t)Z) is convex in t. The first conjecture can be rephrased in the context of completely monotone functions: J(X+√(t)Z) is completely monotone in t. The history of the first conjecture may date back to a problem in mathematical physics studied by McKean in 1966. Apart from these results, we provide a geometrical interpretation to the covariance-preserving transformation and study the concavity of h(√(t)X+√(1-t)Z), revealing its connection with Costa's EPI.