Almost sure convergence in extreme value theory

Let X-1,...,X-n be independent random variables with common distribution function F. Define M-n := max/1 less than or equal to i less than or equal to n X-i and let G(x) be one of the extreme-value distributions. Assume F is an element of D(G), i.e., there exist a(n) > 0 and b(n) is an element of IR such that P{(M-n - b(n))/a(n) less than or equal to x} --> G(x), for x is an element of R. Let 1((-infinity x])(.) denote the indicator function of the set (-infinity,x] and S(G) =: {x : 0 < G(x) < 1}. Obviously, 1((-infinity,x])((M-n - b(n))/a(n)) does not converge almost surely for any x is an element of S(G). But we shall prove P{lim/N-->infinity sup/x is an element of S(G)\1/log N Sigma(n=1)(N) 1/n1((-infinity,x])((M-n - b(n))/a(n)) - G(x)\ = 0} = 1.