Laws of the iterated logarithm for weighted sums of independent random variables
type: Item
instance of: scholarly article
title: Laws of the iterated logarithm for weighted sums of independent random variables
zbMATH Open document ID: 0859.60031
DOI: 10.1016/0167-7152(95)00072-0
published in: Statistics \& Probability Letters
publication date: 1997-04-09T00:00:00Z
review text: Necessary and sufficient conditions for LIL for weighted sums of the form \[ S_n= \sum^n_{k=1} f(k/n)X_k/(2n\log\log n)^{1/2} \] are given. Here \(f\) is a continuous function on \([0,1]\) which can be approximated by polynomials in total variation norm and \(\{X,X_n\}\) is a sequence of i.i.d. mean zero random variables with \(EX^2=1\). The main result states that \(\limsup_{n\to\infty}S_n\) is equal to \(L_2\)-norm of \(f\). Conversely, if this \(\limsup\) is finite for some considered \(f\neq 0\), then \(EX^2<\infty\). Conditions for compact LIL are also given when \(X\) takes values in a separable Banach space, and a LIL for weighted sums is obtained in a Banach space setting.
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 896085
zbMATH Keywords: law of the iterative logarithm
zbMATH Keywords: i.i.d. random variables
zbMATH Keywords: Banach space
zbMATH Keywords: compact law of iterated logarithm
author: Deli Li
author: R. James Tomkins
reviewed by: Andrei Volodin
MaRDI profile type: Publication
full work available at URL: 0167-7152(95)00072-0
OpenAlex ID: W2075080312
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description: scientific article; zbMATH DE number 896085