Fusions of a Probability Distribution

J. Elton, Georgia Institute of Technology - Main Campus
Theodore P. Hill, Georgia Institute of Technology - Main Campus

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Copyright © 1992 Institute of Mathematical Statistics. The definitive version is available at http://dx.doi.org/10.1214/aop/1176989936.

NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.

Abstract

Starting with a Borel probability measure P on X (where X is a separable Banach space or a compact metrizable convex subset of a locally convex topological vector space), the class F(P), called the fusions of P, consists of all Borel probability measures on X which can be obtained from P by fusing parts of the mass of P, that is, by collapsing parts of the mass of P to their respective barycenters. The class F(P) is shown to be convex, and the ordering induced on the space of all Borel probability measures by Q≤ P if and only if Qε F(P) is shown to be transitive and to imply the convex domination ordering. If P has a finite mean, then F(P) is uniformly integrable and Q≤ P is equivalent to Q convexly dominated by P and hence equivalent to the pair (Q, P) being martingalizable. These ideas are applied to obtain new martingale inequalities and a solution to a cost-reward problem concerning optimal fusions of a finite-dimensional distribution.