## The Annals of Applied Probability

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

#### Abstract

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

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

Dates
First available in Project Euclid: 9 August 2018

https://projecteuclid.org/euclid.aoap/1533780277

Digital Object Identifier
doi:10.1214/17-AAP1361

Mathematical Reviews number (MathSciNet)
MR3843833

Zentralblatt MATH identifier
06974755

#### Citation

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. https://projecteuclid.org/euclid.aoap/1533780277

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