Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 130915, 10 pages
Research Article

Almost Sure Central Limit Theorem for a Nonstationary Gaussian Sequence

School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China

Received 4 May 2010; Revised 7 July 2010; Accepted 12 August 2010

Academic Editor: Soo Hak Sung

Copyright © 2010 Qing-pei Zang. 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.


Let {Xn;  n1} be a standardized non-stationary Gaussian sequence, and let denote Sn=k=1nXk, σn=Var(Sn). Under some additional condition, let the constants {uni;  1in, n1} satisfy i=1n(1-Φ(uni))τ as n for some τ0 and min1in unic(logn)1/2, for some c>0, then, we have limn(1/logn)k=1n(1/k)I{i=1k(Xiuki),Sk/σkx}=e-τΦ(x) almost surely for any xR, where I(A) is the indicator function of the event A and Φ(x) stands for the standard normal distribution function.