A note on the law of iterated logarithm for weighted sums of random variables
type: Item
instance of: scholarly article
title: A note on the law of iterated logarithm for weighted sums of random variables
zbMATH Open document ID: 0537.60023
author: Ulrich Stadtmüller
published in: The Annals of Probability
publication date: 1984-01-01T00:00:00Z
review text: We give sufficient conditions for the upper and lower inequality in the law of iterated logarithm for weighted sums \(W_ n=\sum_{k}a_{nk}X_ k\) of i.i.d. random variables. The main condition for the upper inequality is: \(Cor(W_ n,W_ m)\to 1\) as n,\(m\to \infty\), s.t. n and m are close to each other; whereas for the lower inequality we demand that the new r.v.s in \(W_{n+\ell}^ w.\)r. to \(W_ n\) should receive enough weight. In addition the variation within the rows \((a_{nk})\) should not be too large. The proof uses explicit calculations for normal r.v.s and strong approximation results for sums of r.v.s.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 40C05
zbMATH DE Number: 3854099
zbMATH Keywords: law of iterated logarithm
zbMATH Keywords: weighted sums
zbMATH Keywords: strong approximation for sums
zbMATH Keywords: summability matrix
zbMATH Keywords: Borel property
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176993372
OpenAlex ID: W2011895372
DOI: 10.1214/AOP/1176993372
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description: scientific article; zbMATH DE number 3854099