Almost Sure Central Limit Theorem Of Partial Sum And Maximum, And Pickands-type Estimator Of Moving Average Time Series

In the first part of this paper, almost sure central limit theorem of Partial Sum and Maximum was analyzed, and derived them in i.i.d. and α-mixing conditions respectively. The main results are:Theorem A Let {X_n,n ≥ 1} be i.i.d. random variables with nonde-generate common distribution function F, satisfying EX = 0, EX² = 1 and EX^2+δ < ∞, δ > 0. Let S_n = X_i M_n = max{X_i, 1 ≤ i ≤ n}. If there exists constants α_n > 0, b_n ∈ R, such thatwith H(x,∞) = Φ(x), the standard normal distribution, and H(∞,y) is one of the extreme value distributions. Then for any x and y, we haveTheorem B Suppose {X_n,n≥ 1} be a strictly stationary α-mixing sequence of random variables with EX₁ = 0 and mixing coefficient α(n) (log n)^-δ/2,δ > 0. let σ_n > 0 satisfying ES_n² = σ_n² and σ_n/σ_k ≥ (n/k)^γ,γ > 0 when n ≥ k. Assume thatfor which 1 ≥ 2k < l. f(x,y) is the bounded function satisfying Lipschitz condition . Then we have...