Almost Sure Central Limit Theorem for a Nonstationary Gaussian Sequence
نویسندگان
چکیده
Let {Xn; n ≥ 1} be a standardized non-stationary Gaussian sequence, and let denote Sn ∑n k 1 Xk , σn √ Var Sn . Under some additional condition, let the constants {uni; 1 ≤ i ≤ n, n ≥ 1} satisfy ∑n i 1 1−Φ uni → τ as n → ∞ for some τ ≥ 0 and min1≤i≤n uni ≥ c logn , for some c > 0, then, we have limn→∞ 1/ logn ∑n k 1 1/k I{∩i 1 Xi ≤ uki , Sk/σk ≤ x} e−τΦ x almost surely for any x ∈ R, where I A is the indicator function of the eventA andΦ x stands for the standard normal distribution function.
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