Glivenko-Cantelli Theory, Ornstein-Weiss quasi-tilings, and uniform Ergodic Theorems for distribution-valued fields over amenable groups

TL;DR

This paper develops a uniform ergodic theorem for distribution-valued random fields over amenable groups, utilizing quasi-tilings to establish convergence and error bounds in the Banach space setting.

Contribution

It introduces a new ergodic theorem for distribution-valued fields over amenable groups using quasi-tilings, with explicit error estimates.

Findings

Uniform convergence of averages along Følner sequences

Explicit error bounds in the sup norm

Extension of ergodic theorems to distribution-valued fields

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.