Lower Bounds on Multivariate Higher Order Derivatives of Differential Entropy †

This paper studies the properties of the derivatives of differential entropy [Formula: see text] in Costa’s entropy power inequality. For real-valued random variables, Cheng and Geng conjectured that for [Formula: see text] , [Formula: see text] , while McKean conjectured a stronger statement, whereby [Formula: see text]. Here, we study the higher dimensional analogues of these conjectures. In particular, we study the veracity of the following two statements: [Formula: see text] , where n denotes that [Formula: see text] is a random vector taking values in [Formula: see text] , and similarly, [Formula: see text]. In this paper, we prove some new multivariate cases: [Formula: see text]. Motivated by our results, we further propose a weaker version of McKean’s conjecture [Formula: see text] , which is implied by [Formula: see text] and implies [Formula: see text]. We prove some multivariate cases of this conjecture under the log-concave condition: [Formula: see text] and [Formula: see text]. A systematic procedure to prove [Formula: see text] is proposed based on symbolic computation and semidefinite programming, and all the new results mentioned above are explicitly and strictly proved using this procedure.