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August 2018 Glivenko–Cantelli theory, Ornstein–Weiss quasi-tilings, and uniform ergodic theorems for distribution-valued fields over amenable groups
Christoph Schumacher, Fabian Schwarzenberger, Ivan Veselić
Ann. Appl. Probab. 28(4): 2417-2450 (August 2018). DOI: 10.1214/17-AAP1361

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.

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Christoph Schumacher. Fabian Schwarzenberger. Ivan Veselić. "Glivenko–Cantelli theory, Ornstein–Weiss quasi-tilings, and uniform ergodic theorems for distribution-valued fields over amenable groups." Ann. Appl. Probab. 28 (4) 2417 - 2450, August 2018. https://doi.org/10.1214/17-AAP1361

Information

Received: 1 June 2017; Published: August 2018
First available in Project Euclid: 9 August 2018

zbMATH: 06974755
MathSciNet: MR3843833
Digital Object Identifier: 10.1214/17-AAP1361

Subjects:
Primary: 60B12 , 60F99 , 60K35 , 62E20

Keywords: ‎amenable group , empirical measures , Følner sequence , Glivenko–Cantelli theory , quasi-tilings , Uniform convergence

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 4 • August 2018
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