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



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).


  1. 1.
    Afsari, B.: Riemannian \(L^p\) center of mass: existence, uniqueness, and convexity. Proc. Am. Math. Soc. 139(2), 655–673 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics. ETH Zürich, Birkhäuser Verlag, Basel (2005)MATHGoogle Scholar
  3. 3.
    Bačák, M.: The proximal point algorithm in metric spaces. Isr. J. Math. 194(2), 689–701 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bačák, M.: Computing medians and means in Hadamard spaces. SIAM J. Optim. 24(3), 1542–1566 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dudley, R.M.: Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge (2002)Google Scholar
  6. 6.
    Etemadi, N.: Convergence of weighted averages of random variables revisited. Proc. Am. Math. Soc. 134(9), 2739–2744 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Funano, K.: Rate of convergence of stochastic processes with values in \(\mathbb{R}\)-trees and Hadamard manifolds. Osaka J. Math. 47(4), 911–920 (2010)MathSciNetMATHGoogle Scholar
  8. 8.
    Jost, J.: Equilibrium maps between metric spaces. Calc. Var. Partial Differ. Equ. 2(2), 173–204 (1994)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jost, J.: Generalized Harmonic Maps Between Metric Spaces. Geometric Analysis and the Calculus of Variations, pp. 143–174. International Press, Cambridge (1996)MATHGoogle Scholar
  10. 10.
    Kendall, W.: Convexity and the hemisphere. J. Lond. Math. Soc. (2) 43(3), 567–576 (1991)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lim, Y., Pálfia, M.: Weighted deterministic walks for the squares mean on Hadamard spaces. Bull. Lond. Math. Soc. 46(3), 561–570 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Navas, A.: An \(L^1\) ergodic theorem with values in a non-positively curved space via a canonical barycenter map. Ergodic Theory Dyn. Syst. 33(2), 609–623 (2013)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ohta, S.-I.: Convexities of metric spaces. Geom. Dedic. 125, 225–250 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ohta, S.-I., Pálfia, M.: Discrete-time gradient flows and law of large numbers in Alexandrov spaces. Calc. Var. Partial Differ. Equ. 54(2), 1591–1610 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ohta, S.-I., Pálfia, M.: Gradient flows and a Trotter-Kato formula of semi-convex functions on CAT(1)-spaces. Am. J. Math. 139, 937–965 (2017)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Pálfia, M.: Means in metric spaces and the center of mass. J. Math. Anal. Appl. 381(1), 383–391 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Sturm, K.-T.: Nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature. Ann. Probab. 30(3), 1195–1222 (2002)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Sturm, K.-T.: Probability measures on metric spaces of nonpositive curvature. In: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), pp. 357–390. Contemporary Mathematics, vol 338. AMS (2003)Google Scholar
  19. 19.
    Yokota, T.: Convex functions and barycenter on CAT(1)-spaces of small radii. J. Math. Soc. Jpn. 68(3), 1297–1323 (2016)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Yokota, T.: Convex functions and \(p\)-barycenter on CAT(1)-spaces of small radii. Tsukuba J. Math. 41(1), 43–80 (2017)MathSciNetCrossRefMATHGoogle Scholar

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

Authors and Affiliations

  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

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