The Annals of Applied Probability

Glivenko–Cantelli theory, Ornstein–Weiss quasi-tilings, and uniform ergodic theorems for distribution-valued fields over amenable groups

Christoph Schumacher, Fabian Schwarzenberger, and Ivan Veselić

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider random fields indexed by finite subsets of an amenable discrete group, taking values in the Banach-space of bounded right-continuous functions. The field is assumed to be equivariant, local, coordinate-wise monotone and almost additive, with finite range dependence. Using the theory of quasi-tilings we prove an uniform ergodic theorem, more precisely, that averages along a Følner sequence converge uniformly to a limiting function. Moreover, we give explicit error estimates for the approximation in the sup norm.

Article information

Ann. Appl. Probab., Volume 28, Number 4 (2018), 2417-2450.

Received: June 2017
First available in Project Euclid: 9 August 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F99: None of the above, but in this section 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 62E20: Asymptotic distribution theory 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Følner sequence amenable group quasi-tilings Glivenko–Cantelli theory Uniform convergence Empirical measures


Schumacher, Christoph; Schwarzenberger, Fabian; Veselić, Ivan. Glivenko–Cantelli theory, Ornstein–Weiss quasi-tilings, and uniform ergodic theorems for distribution-valued fields over amenable groups. Ann. Appl. Probab. 28 (2018), no. 4, 2417--2450. doi:10.1214/17-AAP1361.

Export citation


  • [1] Adachi, T. (1993). A note on the Følner condition for amenability. Nagoya Math. J. 131 67–74.
  • [2] Aliprantis, C. D. and Border, K. C. (2006). Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd ed. Springer, Berlin.
  • [3] Dehardt, J. (1971). Generalizations of the Glivenko–Cantelli theorem. Ann. Math. Stat. 42 2050–2055.
  • [4] Klenke, A. (2008). Probability Theory. Number 223 in Universitext, 1st ed. Springer, London.
  • [5] Krieger, F. (2009). Le lemme d’Ornstein–Weiss d’aprés Gromov. Dyn., Ergod. Theory Geom. 54 99–112.
  • [6] Lenz, D., Müller, P. and Veselić, I. (2008). Uniform existence of the integrated density of states for models on $\mathbb{Z}^{d}$. Positivity 12 571–589.
  • [7] Lenz, D., Schwarzenberger, F. and Veselić, I. (2010). A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states. Geom. Dedicata 150 1–34.
  • [8] Lenz, D. and Stollmann, P. (2005). An ergodic theorem for Delone dynamical systems and existence of the integrated density of states. J. Anal. Math. 97 1–24.
  • [9] Lenz, D. and Veselić, I. (2009). Hamiltonians on discrete structures: Jumps of the integrated density of states and uniform convergence. Math. Z. 263 813–835.
  • [10] Ornstein, D. and Weiss, B. (1987). Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 1–141.
  • [11] Pogorzelski, F. and Schwarzenberger, F. (2016). A Banach space-valued ergodic theorem for amenable groups and applications. J. Anal. Math. 130 19–69.
  • [12] Schumacher, C., Schwarzenberger, F. and Veselić, I. (2017). A Glivenko–Cantelli theorem for almost additive functions on lattices. Stochastic Process. Appl. 127 179–208.
  • [13] Schwarzenberger, F. (2013). The integrated density of states for operators on groups. Ph.D. thesis, Technische Universität Chemnitz. Available at
  • [14] Tao, T. (2015). Failure of the $L^{1}$ pointwise and maximal ergodic theorems for the free group. Forum Math. Sigma 3 19.
  • [15] Veselić, I. (2005). Spectral analysis of percolation Hamiltonians. Math. Ann. 331 841–865.
  • [16] Weiss, B. (2001). Monotileable amenable groups. In Topology, Ergodic Theory, Real Algebraic Geometry. Amer. Math. Soc. Transl. Ser. 2 202 257–262. Amer. Math. Soc., Providence, RI.
  • [17] Wright, F. T. (1981). The empirical discrepancy over lower layers and a related law of large numbers. Ann. Probab. 9 323–329.