Almost Sure Central Limit Theorem for a Nonstationary Gaussian Sequence

Let {X-n; n >= 1} be a standardized non- stationary Gaussian sequence, and let denote S-n = Sigma(n)(k=1) X-k, sigma(n) = root Var(S-n). Under some additional condition, let the constants {u(ni); 1 <= i <= n, n >= 1} satisfy Sigma(n)(i=1)(1 - Phi(u(ni))) -> tau as n -> infinity for some tau >= 0 and min(1 <= i <= n) u(ni) >= c(log n)(1/2), for some c > 0, then, we have lim(n ->infinity) (1/log n) Sigma(n)(k=1) (1/k)I{boolean AND(k)(i=1) (X-i <= u(ki)), S-k/sigma(k) <= x} = e(-tau)Phi(x) almost surely for any x is an element of R, where I (A) is the indicator function of the event A and Phi(x) stands for the standard normal distribution function.