  In this paper, based on the Fr{\'e}chet mean, we define a notion of
barycenter corresponding to a usual notion of statistical mean. We prove the
existence of Wasserstein barycenters of random distributions 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
distributions or of a growing set of distributions.
