The law of large numbers for partial sum processes indexed by sets
type: Item
instance of: scholarly article
title: The law of large numbers for partial sum processes indexed by sets
zbMATH Open document ID: 0617.60026
author: Joel Zinn
author: Evarist Giné
published in: The Annals of Probability
publication date: 1987-01-01T00:00:00Z
review text: Let \(\{X_ j:\) \(j\in {\mathbb{N}}^ d\}\) be i.i.d. with \(EX_ j=0\) and for subsets A of \([0,1]^ d\) define \(S_ n(A)=\sum_{| j| \leq n}X_ j I(j/n\in A)\). The problem of convergence of \(S_ n(A)\) to 0 uniformly for A in a certain given class \({\mathcal A}\) is studied. It is shown that this property is independent of the particular distribution of the \(X_ j\), and necessary and sufficient conditions are given in terms of the size of \({\mathcal A}\) measured by metric entropy.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 4001093
zbMATH Keywords: law of large numbers
zbMATH Keywords: multi-parameter sum
zbMATH Keywords: uniformity
zbMATH Keywords: metric entropy
reviewed by: Søren Asmussen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992261
OpenAlex ID: W2007610669
DOI: 10.1214/AOP/1176992261
description: scientific article; zbMATH DE number 4001093