Fusions of a Probability Distribution

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 $\mathscr{F}(P)$, 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 $\mathscr{F}(P)$ is shown to be convex, and the ordering induced on the space of all Borel probability measures by $Q \prec P$ if and only if $Q \in \mathscr{F}(P)$ is shown to be transitive and to imply the convex domination ordering. If $P$ has a finite mean, then $\mathscr{F}(P)$ is uniformly integrable and $Q \prec P$ 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.