Existence and consistency of Wasserstein barycenters

@article{LeGouic2015ExistenceAC,
  title={Existence and consistency of Wasserstein barycenters},
  author={Thibaut Le Gouic and Jean-Michel Loubes},
  journal={Probability Theory and Related Fields},
  year={2015},
  volume={168},
  pages={901-917}
}
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 (E, d). 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. 
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