| This thesis is composed of almost surely central limit theorem on the maxima of weak dependent nonstationary Gaussian vector sequence under some conditions, and the almost surely local central limit theorems of the maximum of independent and identically distributed random variables. The main results are:Theorem A Suppsc X1,X2,…be standardized nonstationary Gaussian d dimensional random vectors satisfying (2.1) and (2.2). Let be constants such that, and let Uni(P),P = 1,…,d such thatλn(p)≥c(logn)1/2 for some c＞0.ThenTheorem B Suppse X1, X2,…be standardized nonstationary Gaussian d dimensional random vectors satisfying (2.1), (2.2) and (2.3). Letλn(p)= Uni(p) be constants such that n(1 -Φ(λn(p))) is bounded. ThenTheorem C Suppse X1, X2,…be standardized nonstationary Gaussian d dimensional random vectors satisfying (2.1) and (2.2). Letλn(p) be constants such that n(1 -Φ(λn(p)))→τp as n→∞for someτ-≥O. ThenThorem D Suppse X1,X2,…be standardized nonstationary Gaussian d dimensionalrandomveetorswithδ= ma 1, and (2.5),(2.9) hold for some n→∞for someτp≥0, p = 1,…, d, thenTheorem E Suppse X1, X2,.. be d dimensional Gaussian random vectors with Yn = Xn+mn where satisfy Theorem 2.1.1 and mn, satisfy (2.6) and (2.7) respectively. Ifλn(p)≥c(logn)1/2 for some c＞0, then whereTheorem F Suppse X1, X2,…be d dimensions random variables with Yn = Xn + mn where satisfy Theorem 2. 1. 2 and mn和satisfy (2. 6) and (2. 7) respectively. If (2. 8) holds for some D＞0, then Theorem G Suppse X1, X2,…be standardized d dimensional nonstationary Gaussian random vectors withδ=1, and (2.5),(2.9) hold for some as for some O≤τp,ηp＜∞, thenTheorem H Let X1, X2,…be independent identically distributed random variables with EXi = 0, i = 1,2,….{un}, {un}. If F(un) - F(un)＞andn(1 - F(un)) is bounded as Un＜bn＜un, thcnwhere... |
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