A Note on Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Sums

<p/> <p>Let <inline-formula> <graphic file="1029-242X-2010-234964-i1.gif"/></inline-formula> be a sequence of independent and identically distributed (i.i.d.) random variables and denote <inline-formula> <graphic file="1029-242X-2010-234964-i2.gi...

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Main Authors: Fu Ke-ang, Zang Qing-pei, Wang Zhi-xiang
Format: Article
Language:English
Published: SpringerOpen 2010-01-01
Series:Journal of Inequalities and Applications
Online Access:http://www.journalofinequalitiesandapplications.com/content/2010/234964
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spelling doaj-203ffc06477644899f1dcaf92af2c5e02020-11-25T01:05:30ZengSpringerOpenJournal of Inequalities and Applications1025-58341029-242X2010-01-0120101234964A Note on Almost Sure Central Limit Theorem in the Joint Version for the Maxima and SumsFu Ke-angZang Qing-peiWang Zhi-xiang<p/> <p>Let <inline-formula> <graphic file="1029-242X-2010-234964-i1.gif"/></inline-formula> be a sequence of independent and identically distributed (i.i.d.) random variables and denote <inline-formula> <graphic file="1029-242X-2010-234964-i2.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2010-234964-i3.gif"/></inline-formula>. In this paper, we investigate the almost sure central limit theorem in the joint version for the maxima and sums. If for some numerical sequences <inline-formula> <graphic file="1029-242X-2010-234964-i4.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2010-234964-i5.gif"/></inline-formula> we have <inline-formula> <graphic file="1029-242X-2010-234964-i6.gif"/></inline-formula> for a nondegenerate distribution <inline-formula> <graphic file="1029-242X-2010-234964-i7.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2010-234964-i8.gif"/></inline-formula> is a bounded Lipschitz 1 function, then <inline-formula> <graphic file="1029-242X-2010-234964-i9.gif"/></inline-formula> almost surely, where <inline-formula> <graphic file="1029-242X-2010-234964-i10.gif"/></inline-formula> stands for the standard normal distribution function, <inline-formula> <graphic file="1029-242X-2010-234964-i11.gif"/></inline-formula> ,and <inline-formula> <graphic file="1029-242X-2010-234964-i12.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2010-234964-i13.gif"/></inline-formula>.</p>http://www.journalofinequalitiesandapplications.com/content/2010/234964
collection DOAJ
language English
format Article
sources DOAJ
author Fu Ke-ang
Zang Qing-pei
Wang Zhi-xiang
spellingShingle Fu Ke-ang
Zang Qing-pei
Wang Zhi-xiang
A Note on Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Sums
Journal of Inequalities and Applications
author_facet Fu Ke-ang
Zang Qing-pei
Wang Zhi-xiang
author_sort Fu Ke-ang
title A Note on Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Sums
title_short A Note on Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Sums
title_full A Note on Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Sums
title_fullStr A Note on Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Sums
title_full_unstemmed A Note on Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Sums
title_sort note on almost sure central limit theorem in the joint version for the maxima and sums
publisher SpringerOpen
series Journal of Inequalities and Applications
issn 1025-5834
1029-242X
publishDate 2010-01-01
description <p/> <p>Let <inline-formula> <graphic file="1029-242X-2010-234964-i1.gif"/></inline-formula> be a sequence of independent and identically distributed (i.i.d.) random variables and denote <inline-formula> <graphic file="1029-242X-2010-234964-i2.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2010-234964-i3.gif"/></inline-formula>. In this paper, we investigate the almost sure central limit theorem in the joint version for the maxima and sums. If for some numerical sequences <inline-formula> <graphic file="1029-242X-2010-234964-i4.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2010-234964-i5.gif"/></inline-formula> we have <inline-formula> <graphic file="1029-242X-2010-234964-i6.gif"/></inline-formula> for a nondegenerate distribution <inline-formula> <graphic file="1029-242X-2010-234964-i7.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2010-234964-i8.gif"/></inline-formula> is a bounded Lipschitz 1 function, then <inline-formula> <graphic file="1029-242X-2010-234964-i9.gif"/></inline-formula> almost surely, where <inline-formula> <graphic file="1029-242X-2010-234964-i10.gif"/></inline-formula> stands for the standard normal distribution function, <inline-formula> <graphic file="1029-242X-2010-234964-i11.gif"/></inline-formula> ,and <inline-formula> <graphic file="1029-242X-2010-234964-i12.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2010-234964-i13.gif"/></inline-formula>.</p>
url http://www.journalofinequalitiesandapplications.com/content/2010/234964
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