A Note on Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Sums

Let {X-n; n >= 1} be a sequence of independent and identically distributed (i.i.d.) random variables and denote S-n = Sigma(n)(k=1) X-k, Mn = max(1 <= k <= n)X(k). In this paper, we investigate the almost sure central limit theorem in the joint version for the maxima and sums. If for some numerical sequences (a(n) > 0), (b(n)) we have (M-n - b(n))/a(n) -> G for a nondegenerate distribution G, and f(x, y) is a bounded Lipschitz 1 function, then lim(n ->infinity) (1/D-n) Sigma(n)(k=1) d(k)f(Sk/root k, (M-k - b(k)) /a(k)) = integral integral(infinity)(-infinity)f(x, y) Phi(dx)G (dy) almost surely, where Phi(x) stands for the standard normal distribution function, D-n = Sigma(n)(k=1) d(k), and d(k) = (exp((log k)(alpha)))/k, 0 <= alpha < 1/2.