Relationship between stochastic inequalities and some classical mathematical inequalities

Abstract

The notions of association and dependence of random variables, rearrangements, and heterogeneity via majorization ordering have proven to be most useful for deriving stochastic inequalities. In this survey article we first show that these notions are closely related to three basic inequalities in classical mathematical analysis: Chebyshev's inequality, the Hardy-Littlewood-Pólya rearrangement inequality and Schur functions. We then provide a brief review of some of the recent results in this area. An overall objective is to illustrate that classical mathematical inequalities of this type play a central role in the developments of stochastic inequalities.

Keywords

• stochastic inequalities
• association
• rearrangements
• majorization and Schur functions