Almost Sure Central Limit Theorems Of Extremes From Incomplete Samples

This thesis focuses on the study of the almost sure central limit theorem for some order statistics and the almost sure central limit theorems relcated to complete and incomplete samples of stationary sequences. In the first part, sharp result on almost sure central limit theorem of the first and the second largest maxima is obtained. The main result is as follows.Theorem A Suppose that {X_n} be a sequence of i.i.d. random variables. Let (?) and m_n denote the first and the second maximum of (X₁.....X_n). Assumethat there are normalizing sequences a_n > 0. b_n∈R and a nondegenerate limit distribution G such that (1.1) holds. Assume also that {d_k, k≥1} are positive weights satisfying (1.10)-(1.12). Then for x > y we havewhere H(x,y) = G(y){logG(x) - logG(y) + 1}. G(y) > 0 (and to zero when G(y) = 0).In the second part, almost sure max-limit theorems of complete and incompletesamples of i.i.d. sequence arc analyzed. The related results arc also extended to weak dependent stationary Gaussian sequence. Followings are the main resuts.Theorem B Let {X_n^*} be a sequence of i.i.d. random variables with common distribution function F(x) such that F∈D(G). Suppose the indicator variables {ε_n} are a sequence of independent random variables. which are independent of {X_n^*} and satisfy T_n/n (?) p∈(0.1], where T_n =∑_k=1ⁿε_k .then for all real x3-C₅, if there exist numerical sequences {u_n}.{v_n} such that u_n < v_n;n≥1, and n(1-Φ(u_n))→τ₁∈[0,+∞), n(1-Φ(v_n))→τ₂∈[0.∞) as n→∞. Then we haveFurthermore, for x < ywhere H₁(x,y) = exp(-pe^-x)exp(-(l -p)e^-y).In the third part of this thesis. almost sure central limit theorem of the partial sums and maximums from complete and incomplete samples of i.i.d. sequence is considered. Related results also are extended to weak dependent Gaussian sequence. The main results areTheorem D Suppose that conditions C₁, C₆ and C₇ hold for the i.i.d. sequence {X_n^*}. Furthermore, E(X₁^*)^2+δ <∞for some 0 <δ< 1. Then for all x < y, z∈R we haveTheorem E Let {X_n} be a standardized stationary Gaussian sequence. Assume that conditions G₁ to G₄ hold. Then for all z∈R.Furthermore, for all real x < y, z∈R,...