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

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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1533780277

Digital Object Identifier
doi:10.1214/17-AAP1361

Mathematical Reviews number (MathSciNet)
MR3843833

Zentralblatt MATH identifier
06974755

Subjects
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]

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

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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