Let be independent random variables with and . Set and assume that . We prove that under the Kolmogorov condition we have for any almost everywhere continuous function satisfying , . We also show that replacing the o in (1) by O, relation (2) becomes generally false. Finally, in the case when (1) is not assumed, we give an optimal condition for (2) in terms of the remainder term in the Wiener approximation of the partial sum process by a Wiener process.
Keywords: Almost sure central limit theorem; Sums of independent random variables; Weighted averages.
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