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Hessian orders and multinormal distributions. (English) Zbl 1177.60020

Summary: Several well known integral stochastic orders (like the convex order, the supermodular order, etc.) can be defined in terms of the Hessian matrix of a class of functions. Here we consider a generic Hessian order, i.e., an integral stochastic order defined through a convex cone \(\mathcal H\) of Hessian matrices, and we prove that if two random vectors are ordered by the Hessian order, then their means are equal and the difference of their covariance matrices belongs to the dual of \(\mathcal H\). Then we show that the same conditions are also sufficient for multinormal random vectors. We study several particular cases of this general result.

MSC:

60E15 Inequalities; stochastic orderings
15B48 Positive matrices and their generalizations; cones of matrices
26B25 Convexity of real functions of several variables, generalizations
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