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 QP 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 QP 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.



URL: http://digitalcommons.calpoly.edu/rgp_rsr/39