Fréchet Barycenters and a Law of Large Numbers for Measures on the Real Line

Abstract

Endow the space $\mathcal{P}(\mathbb{R})$ of probability measures on $\mathbb{R}$ with a transportation cost $J(\mu ,\nu )$ generated by a translation-invariant convex cost function. For a probability distribution on $\mathcal{P}(\mathbb{R})$ we formulate a notion of average with respect to this transportation cost, called here the Fréchet barycenter, prove a version of the law of large numbers for Fréchet barycenters, and briefly discuss the structure of $\mathcal{P}(\mathbb{R})$ related to the transportation cost $J$.

15 pages; added results about weak continuity of barycenter, revised arguments in section 5, section 7 removed (partially merged with section 6)