SOME RECENT DEVELOPMENTS ON MAJORIZATION INEQUALITIES IN PROBABILITY AND STATISTICS

We review some recent developments on majorization inequalities in pro bability and statistics. In Section 2 we describe some related general izations, including majorization in an integral form, multivariate maj orization, stochastic majorization, and other related orderings (inclu de the Lorentz ordering and arrangement increasing ordering); and stat e some results. Section 3 concerns partial orderings of probability co ntents of certain classes of geometric regions in R(n) via a majorizat ion ordering of their parameter vectors. Section 4 deals with some app lications of majorization theory to dependence and peakedness ordering s of random variables. In most cases the described results have become available since the publication of Marshall and Olkin's book (1979), and are directly or indirectly inspired by it.