Almost sure convergence in extreme value theory

Let X₁ , . . . , X_n be independent random variables with common distribution function F. Define M_n := max X_i1≤i≤n and let G(cursive Greek chi) be one of the extreme - value distributions. Assume F ∈ D(G), i.e., there exist a_n > 0 and b_n ∈ double-struck R sign such that P{(M_n - b_n)/a_n ≤ cursive Greek chi} → G(cursive Greek chi), for cursive Greek chi ∈ double-struck R sign . Let 1(_{-∞,cursive Greek chi}](·) denote the indicator function of the set (-∞, cursive Greek chi] and S(G) =: {cursive Greek chi : 0 < G(cursive Greek chi) < 1}. Obviously, 1(_{-∞,cursive Greek chi}]((M_n - b_n)/a_n) does not converge almost surely for any cursive Greek chi ∈ S(G). But we shall prove (equation presented).