Probability of random vector hitting in a polyhedral frustum of a cone: Majorization aspect

The article presents conditions under which the probability of a linear combination of random vectors falling into a polyhedral oblate (from above) cone, in particular, into frustum of a cone is a Schur-concave function of the vector corresponding this linear combination. It is required that the oblate cone is convex, it contains the point 0, its edges are parallel to the coordinate axes, and the distribution density of vectors is a logarithmically concave sign-invariant function. In addition, the characterization was obtained in differential form functions that preserve one known pre-order, which is inside the majorization pre-order.