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

• 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
• ISSN: 1025-5834; 1029-242X

<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>