A note on uniform laws of averages for dependent processes
type: Item
instance of: scholarly article
title: A note on uniform laws of averages for dependent processes
zbMATH Open document ID: 0776.60042
DOI: 10.1016/0167-7152(93)90163-D
author: Amir Dembo
author: Andrew B. Nobel
published in: Statistics \& Probability Letters
publication date: 1993-11-24T00:00:00Z
review text: Let \({\mathcal F}\) be a class of real-valued functions defined on a measurable space \(({\mathcal X},{\mathcal S})\) such that \[ \sup_{f\in{\mathcal F}}\left|{1\over n}\sum^{n-1}_{i=0}f(X_ i)-Ef(X_ 0)\right|\to 0 \tag{*} \] almost surely for some \({\mathcal X}\)-valued i.i.d. sequence \(\{X_ n, n\geq 0\}\) with distribution function (d.f.) \(F\). Under certain regularity conditions on \({\mathcal F}\), it is shown that \((*)\) holds for every stationary, absolutely regular stochastic sequence \(\{X_ n,-\infty<n<\infty\}\) whose marginal d.f. is \(F\).
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60F20
zbMATH DE Number: 218534
zbMATH Keywords: uniform convergence
zbMATH Keywords: permissible class
zbMATH Keywords: weak Bernoulli processes
zbMATH Keywords: regularity conditions
reviewed by: R. James Tomkins
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 218534