Convergence Rate Of Conditional Moments, Large Quantile Estimation And Almost Sure Central Limit Theorem On Maxima And Minima

Three parts composes this thesis. Firstly the convergence rates of conditional moments of random variables were discussed as the distribution function belongs to the domains of attraction of D(Φ_α),D(Ψ_α) and -D(Λ) respectively. The main results are:Theorem A If is p - order conditional moments of X, let ,sgn(A0_t) is constant for large t, satisfying (2.4) and (2.5), thenTheorem B If F ,let x₀ = sup{x : F(x) < 1},B₀(t) = p +α, sgn(B₀(t))is constant for large t, satisfying (2.9) and (2.10), thenTheorem C If F ,let P(t) = - log J_p(t) satisfying (2.14), thenwhereIn the second part,we give the large quantile (x_Pn) estimation of fixed smoothing parameter:Theorem D If F∈D(G_γ) (γ> 0),there exists a regularly varying function and lim (t) =∞, such that lim∞locally uniformly for x > 0. m is a fixed constant,np_n→∞, p_n→0 as n→∞, then Theorem E If F∈D(G_γ) (γ< 0),there exists a regularly varying function ,and lim sup locally uniformly for x > 0. m is a fixed constant, np_n→∞,p_n→0 as n→∞,thenTheorem F If F∈D(G_γ) (γ= 0),there exists a regularly varying functionρ(t), lim ,and lim sup for all x > 0, where dy. m is a fixed constant,np_n→c as n→∞,c∈R⁺, thenIn the thrid part, almost sure central limit theorem of maximum and minima on dependent Gaussian sequence was analyzed, which wasTheorem G suppose be a standardized stationary Gaussion sequences with covariance r_n→0 as n→∞,andwhereε> 0,γ> 2. And(1). If there exist constants ofμ_n,v_n and 0≤τ< 00,0≤θ<∞, such that and are bounded,then...