On the Probability of a Linear Combination of Random Vectors Hitting n-Dimensional Parallelepiped

The paper presents conditions under which the probability of a linear combination of independent identically distributed random vectors hitting an n -dimensional rectangular parallelepiped is a Schur-concave function of the vector corresponding to this linear combination. It is required that the parallelepiped contain the point 0, its edges be parallel to the coordinate axes, and the vector distribution density be a logarithmically concave sign-invariant function. In addition, a new pre-order within the majorization is proposed, and a characterization of functions preserving this pre-order is obtained in differential form.