Almost sure central limit theorem for partial sums and maxima

Let X, X-1, X-2,... be i.i.d. random variables with nondegenerate common distribution function F, satisfying EX = 0, EX2 = 1. Let S-n = Sigma(n)(i=1) X-i and M-n = max{X-i, 1 <= i <= n}. Suppose there exists constants a(n) > 0, b(n) is an element of R and a nondegenrate distribution G(y) such that lim(n ->infinity) P (M-n - b(n)/a(n) <= y) = G(y), -infinity < y < +infinity. Then, we have lim(n ->infinity) 1/log n (n)Sigma(k=1) 1/k f (S-k/root k, M-k - b(k)/a(k)) = integral integral f(x,y)Phi(dx)G(dy) almost Surely. where f (x. y) denotes the bounded Lipschitz 1 function and Phi(x) is the standard normal distribution function. (C) 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim