Some limit theorems for the empirical process indexed by functions
type: Item
instance of: scholarly article
title: Some limit theorems for the empirical process indexed by functions
zbMATH Open document ID: 0587.60027
DOI: 10.1007/BF01845640
publication date: 1986-01-01T00:00:00Z
review text: Let \(G_ n\), \(n\geq 1\), be a sequence of classes of real-valued measurable functions defined on a probability space (S,\({\mathcal S},P)\). Under weak metric entropy conditions on \(G_ n\), \(n\geq 1\), and under growth conditions on \(\sup \{\| g\|_{\infty}:\) \(g\in G_ n\}\) and \(\sup \{\| g\|_ 2:\) \(g\in G_ n\}\), we show that there are non-zero numerical constants \(C_ 1\) and \(C_ 2\) such that \[ C_ 1\leq \overline{\lim}_{n\to \infty}\sup_{g\in G_ n}| \alpha (n)\int g(dP_ n-dP)| \leq C_ 2<\infty \quad a.s.\quad (P^{{\mathbb{N}}}), \] where \(\alpha\) (n) is a non-decreasing function of n related to the metric entropy of \(G_ n\). A few applications of this general result are considered: we obtain a.s. rates of uniform convergence for the empirical process indexed by intervals as well as a.s. rates of uniform convergence for the empirical characteristic function over expanding intervals.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G60
zbMATH DE Number: 3942612
zbMATH Keywords: weak metric entropy conditions
zbMATH Keywords: rates of uniform convergence
zbMATH Keywords: empirical process indexed by intervals
zbMATH Keywords: empirical characteristic function
author: J. E. Yukich
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 3942612