Almost Sure Convergence For Sample Range And Joint Asymptotic Distributions Of Exceedances Point Process And Partial Sum

Almost surely central limit theorem on sample range of i. i. d sequence, the asymptotic joint distributions of exceedances point process and the partial sum formed by three kinds of standard Gaussian sequences are the main topics on this thesis. The main results are:Theorem A. Let be i. i. d random variables with sample range R_n =max{X_i, 1≤i≤n} - min{X_i, 1≤i≤n}. If there exist some numerical sequences ak,β_k, k = 1, 2,…, and nondegenerated distribution G, such thatthen almost surely for any continuity points x of G.Theorem B. Let be a standardized stationary Gaussian sequence, and N_n be the exceedances Of random level formed by X₁,X₂,…, X_n. Let r_n = EX₁X_n+1, Under conditions (1.5),(1.6)and (1.7), we have N_n and S_n are asymptotically independent and N_n N,where N is a Poisson process on (0,1].Theorem C. Let be a standardized non-stationary Gaussian sequence, and Q_n be the exceedances of levelμ_n(x) formed by X₁,X₂,…, X_n, Xi, If conditions (1.9),(1.10) are satisfied, then Q_n and S_n are asymptotically independent and Q_n Q, where Q is a Poisson process on (0,1].Theorem D. Let be a standardized strong dependent non-stationary Gaussian sequence, and the exceedances point process G_n be the exceedances of levelμ_n(x) formed by X₁, X₂,…, X_n, S_n= Under conditions (1.11),(1.12), G_n converges in distribution to a Cox-process and as B₁, B₂,…,B_k are disjoint Borel subsets of (0, 1] with m(αBi) = 0, i=1, 2,…, k, where m(·) is Lebesgue measure, l₁, l₂,…, l_k are non-negative integer numbers.Keywords: Almost surely central limit theorem, Sample range, Maxima, k-th Maxima, Partial sum, Poisson process, Cox-process, Stationary Gaussian sequence, Non-stationary Gaussian sequence, Exceedances point process...