Second-order Regular Variation And Application In Rates Of Convergence In Extreme-value

Suppose {X_i, i = 1,2, ...) be an i.i.d random sequence with common distribution function F{x), M_n = max{X₁,...,X_n},n =1,2,.... By Gneaenko,B.V. (1943), If there exist constants an> 0, bn ∈ R and nondegenerate distribution function G(x), such that P(a_n^-1(M_n - b_n) ≤ x) →d G(x), then G(x) is one of the types of G_γ(x), where G_γ(x) = exp{-(1 + γx)^-1/γ},γ ∈R, 1 +γx ≥ 0. Let V := (- log^-1 F)(?), without of loss of generality, supposewhich equivalent to existing function a(t),t ∈ R⁺, such that when x > 0,Lots of authors considered the rate of convergence in extreme-value because of the demand of extreme value applications.Three parts is mainly included in this paper. In the second part of this article, regular varying function, second-order varying function and generalized varying function were introduced and studied. Here the differentiable function f with positive difference f', and if there exist a(t) > 0, A(t),A(t) → 0, as t →∞, A(t) has constant sign near infinity and |A| ∈ Rv_p for p ≤ 0, such thatthen f'(t) satisfies a second-order regular varying condition. As f(t) has this property, we find the suitable a*(t) and A*(t) of a(t) and A(t) respectively. And the main result was obtained, i.e. Proposition 2.6.In the third part, some lemmas are proved when a_n = a*(n),An = A*(n), which are the basis of the main results, such as Lemma 3.3.