GLIVENKO-CANTELLI THEORY, ORNSTEIN-WEISS QUASI-TILINGS, AND UNIFORM ERGODIC THEOREMS FOR DISTRIBUTION-VALUED FIELDS OVER AMENABLE GROUPS

• 成果类型: Article
• 署名作者: Schumacher, Christoph; Schwarzenberger, Fabian; Veselic, Ivan
• 署名单位: Dortmund University of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/17-AAP1361
• 发表日期: 2018
• 页码: 2417-2450
• 关键词: Existence

摘要

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