%0 Journal Article
%T A Note on Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Sums
%A Qing-pei Zang
%A Zhi-xiang Wang
%A Ke-ang Fu
%J Journal of Inequalities and Applications
%D 2010
%I Springer
%R 10.1155/2010/234964
%X Let {Xn;n¡Ý1} be a sequence of independent and identically distributed (i.i.d.) random variables and denote Sn=¡Æk=1nXk, Mn=max 1¡Ük¡Ün 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¡ú  G for a nondegenerate distribution G, and f(x,y) is a bounded Lipschitz 1 function, then lim n¡ú¡Þ (1/Dn)¡Æk=1ndkf(Sk/k,(Mk-bk)/ak)= -¡Þ¡Þf(x,y)¦µ(dx)G(dy) almost surely, where ¦µ(x) stands for the standard normal distribution function, Dn=¡Æk=1ndk ,and dk=(exp ((log k)¦Á))/k, 0¡Ü¦Á<1/2.
%U http://dx.doi.org/10.1155/2010/234964