Local And Global Almost Sure Central Limit Theorems On Extremes

Almost sure local central limit theorems for the maxima of i.i.d random variables and stationary Gaussian sequences, almost sure central limit theorem for max and sums of nonstationary Gaussian sequences are the main topics on this thesis. The main results areTheorem A Let X₁,X₂,... be i.i.d random variables with EX₁ = 0,E|X₁|³ <∞. The marginal distribution function F∈D(G). Let real sequences {v_k,u_k,k≥1} satisfy v_k≤v_k+1≤u_k≤u_k+1. Setting M_k = max_{1≤i≤k}X_i,p_k=P(v_k≤M_k≤u_k) andIf n(1-F(v_n))<∞andThenTheorem B Let X₁,X₂,... be a standardized stationary Gaussian sequence with covariance r_n=cov(X₁,X_n+1). Assume real sequences {u_n,v_n,n≥1} satisfyDenote Thenprovided r_n logn(loglogn)^-(1+ε)=O(1).Theorem C Let X₁,X₂,... be i.i.d random variables with common distribution function F with left endpoint x_l and right endpoint x₀. Suppose F is absolutely continuous with bounded density F'. Let positive sequence {d_n,n≥1} satisfyd_n→∞as n→∞Following almost sure local limit theorems on densities are obtained:(i). If there exists someα> 0, such that lim_{x→∞}xF'(x)/(1- F(x)) =α, forany h > 0 and x∈(0, +∞), we have(ii). If there exists someα> 0, such that lim_x↑x0(x₀-x)F'(x)/(1- F(x)) =α.For any h > 0 and x∈(-∞,0), we have(iii)．If lim_x↑x0F'(x)integral from n=x to x₀ (1-F(t))dt／(1-F(x))²=1 holds,then for any h>0and x∈R．we haveTheorem D Let X₁,X₂,...be a standardized nonstationary Gaussian sequences．Assume some numerical sequencesρ_n and{u_ni,1≤i≤n,i=1,2,...}satisfy (?)ρ_n<δ<1,n(1-Φ(λ_n))is bounded(whereλ_n=(?)u_ni)and(?)(1-Φ(u_ni))→(?) for some (?)<∞．If thenfor all y∈(-∞,∞).