Law of large numbers in CAT(1)-spaces of small radii

  • Takumi Yokota


We prove the law of large numbers formulated with barycenter of probability measures for random variables with values in small balls in CAT(1)-spaces. This extends the previous results in CAT(0)-spaces and CAT(1)-spaces of small diameters by Sturm and Ohta–Pálfia.

Mathematics Subject Classification




The author is grateful to Professor Kazuhiro Kuwae for drawing his attention to Kendall’s convex function and Professor Emanuel Milman for suggesting our Theorem 2. He also thanks the referee for the thorough reading of the manuscript. This work was partly supported by JSPS KAKENHI (No. 26800035).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

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