Fr\'echet Barycenters and a Law of Large Numbers for Measures on the Real Line

TL;DR

This paper introduces the concept of Fréchet barycenters in the space of probability measures on the real line, establishes a law of large numbers for these barycenters, and explores the structure of this measure space under a transportation cost.

Contribution

It formulates a new notion of averaging probability measures via transportation costs and proves a law of large numbers for these averages, advancing the understanding of measure spaces.

Findings

Defined Fréchet barycenters with respect to a transportation cost

Proved a law of large numbers for Fréchet barycenters

Analyzed the structure of the measure space under the transportation cost

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 Fr\'echet barycenter, prove a version of the law of large numbers for Fr\'echet barycenters, and briefly discuss the structure of $\mathcal{P}(\mathbb{R})$ related to the transportation cost $J$.