Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 234964, 7 pages
doi:10.1155/2010/234964
Research Article

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

Qing-pei Zang,1 Zhi-xiang Wang,1 and Ke-ang Fu2

1School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China
2School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China

Received 30 March 2010; Accepted 25 May 2010

Academic Editor: Jewgeni Dshalalow

Copyright © 2010 Qing-pei Zang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let {Xn;n≥1} be a sequence of independent and identically distributed (i.i.d.) random variables and denote Sn=∑k=1nXk, Mn=max⁡1≤k≤n⁡Xk. 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 (an>0), (bn) we have (Mn-bn)/an→𝒟G for a nondegenerate distribution G, and f(x,y) is a bounded Lipschitz 1 function, then lim⁡n→∞⁡(1/Dn)∑k=1ndkf(Sk/k,(Mk-bk)/ak)=∬-∞∞f(x,y)Φ(dx)G(dy) almost surely, where Φ(x) stands for the standard normal distribution function, Dn=∑k=1ndk ,and dk=(exp⁡((log⁡k)α))/k, 0≤α<1/2.