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 Foelner sequence converge uniformly to a limiting function. Moreover we give explicit error estimates for the approximation in the sup norm.

Locations

• arXiv (Cornell University) - View - PDF
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• The Annals of Applied Probability - View - PDF