A general form of the law of the iterated logarithm for the weighted multivariate empirical process
type: Item
instance of: scholarly article
title: A general form of the law of the iterated logarithm for the weighted multivariate empirical process
zbMATH Open document ID: 0609.60037
DOI: 10.1016/0167-7152(87)90035-6
author: John H. J. Einmahl
published in: Statistics \& Probability Letters
publication date: 1987-01-01T00:00:00Z
full work available at URL: JE28 .PDF
review text: Consider a sequence of i.i.d. random vectors \(X_ 1,X_ 2,..\). which are uniformly distributed on \([0,1]^ d\), \(d\geq 1\). For \(t=(t_ 1,...,t_ d)\), let \(| t| =\prod^{d}_{i=1}t_ i\). For \(c>0\) and \(n\geq 1\) let \(a_ n=c \log \log n/n\) and let \(F_ n(t)\) be the empirical distribution function of \(X_ 1,...,X_ n\). The main result establishes the almost sure value of \[ \limsup_{n\to \infty}\sup_{a_ n\leq | t| \leq 1}n^{\nu}| F_ n(t)- | t| | /((\log \log n)^{\nu}| t|^{1-\nu}) \] where \(0\leq \nu \leq 1/2\). The author indicates that this theorem contains all known results of this type.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F99
zbMATH DE Number: 3986290
zbMATH Keywords: exponential probability inequalities
zbMATH Keywords: independent random vectors
zbMATH Keywords: weighted empirical process
zbMATH Keywords: law of the iterated logarithm
zbMATH Keywords: empirical distribution function
reviewed by: R. James Tomkins
MaRDI profile type: Publication
OpenAlex ID: W2094594969
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description: scientific article; zbMATH DE number 3986290