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Working paper

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

math. arxive. Cornell University, 2015. No. 1512.08421.
Kroshnin A., Sobolevski A.
Endow the space P(R) of probability measures on R with a transportation cost J(mu, nu) generated by a translation-invariant convex cost function. For a probability distribution on P(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 discuss the structure of P(R) related to the transportation cost J.