| Abstract |
Let H(X_t) be the differential entropy of an n-dimensional random vector X_t introduced by Costa. Cheng and Geng conjectured that C_1(m,n): (-1)^{m+1}(\d^m/\d^m t)H(X_t)\ge0. McKean conjectured that 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 conjecture 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) under the log-concave condition. A systematic procedure to prove C_l(m,n) is proposed and the results mentioned above are proved using this procedure.
|
