We prove central limit theorems and related asymptotic results for where W is a Wiener process and Sk are partial sums of i.i.d. random variables with mean 0 and variance 1. The integrability and smoothness conditions made on f are optimal in a number of important cases.
BERKES, I. and HORVÁTH, L., almost sure invariance principles for logarithmic averages, Studia Sci. Math. Hungar.33 (1997), 1-24. MR98f:60054