Probability Theory and Related Fields

, Volume 168, Issue 3–4, pp 901–917 | Cite as

Existence and consistency of Wasserstein barycenters

  • Thibaut Le GouicEmail author
  • Jean-Michel Loubes


Based on the Fréchet mean, we define a notion of barycenter corresponding to a usual notion of statistical mean. We prove the existence of Wasserstein barycenters of random probabilities defined on a geodesic space (Ed). We also prove the consistency of this barycenter in a general setting, that includes taking barycenters of empirical versions of the probability measures or of a growing set of probability measures.


Fréchet mean Wasserstein distance Barycenter Existence Consistency 

Mathematics Subject Classification

60D05 62H10 62G20 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373MarseilleFrance
  2. 2.Université de Toulouse, Institut de Mathématiques de ToulouseToulouseFrance

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