On the convergence of generalized moments in almost sure central limit theorem

Let {zeta(k)} be the normalized sums corresponding to a sequence of i.i.d. variables with zero mean and unit variance, Define random measures Q(n) = 1/log n (k=1)Sigma(n) 1/k delta(zeta k) and let G be the normal distribution. We show that for each continuous function h satisfying integral h d G < infinity and a mild regularity assumption, one has lim(n-->infinity) integral h d Q(n) = integral h dG a.s. (C) 1998 Elsevier Science B.V. All rights reserved. AMS classification: primary 60F15; secondary 60F05; 60F17; 60B12.