Probability and Expectation Inequalities

This paper introduces a mathematical framework within which a wide variety of known and new inequalities can be viewed from a common perspective. Probability and expectation inequalities of the following types are considered: (a) P(ZEA) ~ P(Z'EA) for some class of sets A, (b) Ek(Z) ~ Ek(Z') for some class of functions k, and (c) Et(Z) ~ Ek(Z') for some class of pairs of functions t and k. It is shown, sometimes using explicit constructions of Z and Z', that, in several cases, (a) <=> (b) <=> (c); included here are cases of normal and elliptically contoured distributions. A case where (a) ~> (b) <~> (c) is studied and is expressed in terms of "n-monotone" functions for (some of) which integral representations are obtained. Also, necessary and sufficient conditions for (c) are given.