Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics

@article{AhidarCoutrix2019ConvergenceRF,
  title={Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics},
  author={Adil Ahidar-Coutrix and Thibaut Le Gouic and Quentin Paris},
  journal={Probability Theory and Related Fields},
  year={2019},
  volume={177},
  pages={323-368}
}
This paper provides rates of convergence for empirical (generalised) barycenters on compact geodesic metric spaces under general conditions using empirical processes techniques. Our main assumption is termed a variance inequality and provides a strong connection between usual assumptions in the field of empirical processes and central concepts of metric geometry. We study the validity of variance inequalities in spaces of non-positive and non-negative Aleksandrov curvature. In this last… 
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