Lower Bound on Derivatives of Costa's Differential Entropy

Abstract

Several conjectures concern the lower bound for the differential entropy $H(X_t)$ of an $n$-dimensional random vector $X_t$ introduced by Costa. Cheng and Geng conjectured that $H(X_t)$ is completely monotone, that is, $C_1(m,n): (-1)^{m+1}(d^m/d^m t)H(X_t)\ge0$. McKean conjectured that Gaussian $X_{Gt}$ achieves the minimum of $(-1)^{m+1}(d^m/d^m t)H(X_t)$ under certain conditions, that is, $C_2(m,n): (-1)^{m+1}(d^m/d^m t)H(X_t)\ge(-1)^{m+1}(d^m/d^m t)H(X_{Gt})$. McKean's conjecture was only considered in the univariate case before: $C_2(1,1)$ and $C_2(2,1)$ were proved by McKean and $C_2(i,1),i=3,4,5$ were proved by Zhang-Anantharam-Geng under the log-concave condition. In this paper, we prove $C_2(1,n)$, $C_2(2,n)$ and observe that McKean's conjecture might not be true for $n>1$ and $m>2$. We further propose a weaker version $C_3(m,n): (-1)^{m+1}(d^m/d^m t)H(X_t)\ge(-1)^{m+1}\frac{1}{n}(d^m/d^m t)H(X_{Gt})$ and prove $C_3(3,2)$, $C_3(3,3)$, $C_3(3,4)$, $C_3(4,2)$ under the log-concave condition. A systematical procedure to prove $C_l(m,n)$ is proposed based on semidefinite programming and the results mentioned above are proved using this procedure.