On the law of the iterated logarithm for martingales
type: Item
instance of: scholarly article
title: On the law of the iterated logarithm for martingales
zbMATH Open document ID: 0758.60026
author: Evan Fisher
published in: The Annals of Probability
publication date: 1993-01-16T00:00:00Z
review text: Let \(\{U_ n\), \(n\geq 1\}\) be a martingale adapted to an increasing sequence of \(\sigma\)-fields \(\{{\mathcal F}_ n\), \(n\geq 1\}\) and \(s_ n^ 2=\sum_{i=1}^ n E(X_ i^ 2\mid{\mathcal F}_{i-1})\), \(n\geq 1\). The main result establishes (with probability 1) an upper bound of \(\limsup_{n\to\infty}U_ n/s_ n\varphi(s_ n)\) with \(\varphi(x):=(2\log\log(x^ 2\vee e^ 2))\) which generalizes the corresponding estimate for sums of i.i.d. r.v.'s derived by \textit{V. A. Egorov} [Theory Probab. Appl. 14, 693-699 (1969); translation from Teor. Veroyatn. Primen. 14, 722-729 (1969; Zbl 0211.489)].
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G42
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 89783
zbMATH Keywords: law of iterated logarithm
zbMATH Keywords: discrete-time martingales
reviewed by: Lothar Heinrich
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176989799
OpenAlex ID: W2069044858
DOI: 10.1214/AOP/1176989799
description: scientific article; zbMATH DE number 89783