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

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 \(J\).