Ordering of Distributions and Rearrangement of Functions

Abstract

Some characterizations of semiorders defined on the set of all probability measures on $R^n$ by the set of Schur-convex functions and by some subsets of all convex functions are proved. A connection of these results to the theorem of Hardy, Littlewood and Polya on the rearrangement of functions is discussed. Furthermore, by means of the results on the ordering of probability measures a generalization of a theorem on doubly stochastic linear operators due to Ryff is proved.