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.
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Marshall, A. W., Olkin, I., & Arnold, B. C. (2011). Inequalities: Theory of majorization and its applications (2nd ed.). New York: Springer.
Nevius, S. E., Proschan, F., & Sethuraman, J. (1977). Schur functions in statistics, II. Stochastic majorization. Annals of Statistics, 5, 263–273.
Strassen, V. (1965). The existence of probability measures with given marginals. Annals of Mathematical Statistics, 36, 423–439.
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Arnold, B.C., Sarabia, J.M. (2018). Stochastic Majorization. In: Majorization and the Lorenz Order with Applications in Applied Mathematics and Economics. Statistics for Social and Behavioral Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-93773-1_8
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DOI: https://doi.org/10.1007/978-3-319-93773-1_8
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