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.

Abstract

Let {Xn;  n≥1} be a standardized non-stationary Gaussian sequence, and let denote Sn=∑k=1nXk, σn=Var(Sn). Under some additional condition, let the constants {uni;  1≤i≤n, n≥1} satisfy ∑i=1n(1-Φ(uni))→τ as n→∞ for some τ≥0 and min1≤i≤n uni≥c(logn)1/2, for some c>0, then, we have limn→∞(1/logn)∑k=1n(1/k)I{∩i=1k(Xi≤uki),Sk/σk≤x}=e-τΦ(x) almost surely for any x∈R, where I(A) is the indicator function of the event A and Φ(x) stands for the standard normal distribution function.