Stochastic Majorization

Abstract

We return to the original setting in which Hardy, Littlewood, and Polya discussed majorization, i.e., vectors in $\mathbb {R}^n$. However, we now consider random variables which take on values in $\mathbb {R}^n$. If $\underline {X}$ and $\underline {Y}$ are such n-dimensional random variables, then for given realizations of $\underline {X}$ and $\underline {Y}$, say $\underline {x}$ and $\underline {y}$, we may or may not have $\underline {x}\le _M \underline {y}$. If we have such a relation for every realization of $( \underline {X}, \underline {Y})$, then we have a very strong version of stochastic majorization holding between $\underline {X}$ and $\underline {Y}$. However, we may be interested in weaker versions. Certainly, for most purposes $P( \underline {X}\le _M \underline {Y})=1$ would be more than adequate. We will content ourselves with the requirement that there be versions of $\underline {X}$ and $\underline {Y}$ for which $\underline {X}$ is almost surely majorized by $\underline {Y}$. However, this will not be transparent from the Definition (8.1.2 below). We will focus attention on one particular form of stochastic majorization, the one proposed by Nevius et al. (Ann Stat 5:263–273 (1977)). We will mention in passing other possible definitions and refer the interested reader to the rather complicated diagram on page 426 of Marshall et al. (Inequalities: theory of majorization and its applications, 2nd edn. Springer, New York (2011)), which summarizes known facts about the interrelationships between the various brands of stochastic majorization.