Summary
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(ZεA)≧ P(Z′εA) for some class of setsA, (b)ℰk(Z)≧ℰk(Z′) for some class of functionsk, and (c)ℰl(Z)≧ℰk(Z′) for some class of pairs of functionsl andk. It is shown, sometimes using explicit constructions ofZ andZ′, 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.
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Research supported by the Air Force Office of Scientific Research under Grants AFOSR-75-2796 and AFOSR-80-0080
Research supported by the National Science Foundation under Grants MCS78-01240 and MCS81-00748
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Cambanis, S., Simons, G. Probability and expectation inequalities. Z. Wahrscheinlichkeitstheorie verw Gebiete 59, 1–25 (1982). https://doi.org/10.1007/BF00575522
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DOI: https://doi.org/10.1007/BF00575522