Lower Bound on Derivatives of Costa's Differential Entropy
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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)≥0. 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)≥(-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)≥(-1)^m+11/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.
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