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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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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