# Almost sure central limit theory for products of sums of partial sums

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## Abstract

Consider a sequence of i.i.d. positive random variables. An universal result in almost sure limit theorem for products of sums of partial sums is established. We will show that the almost sure limit theorem holds under a fairly general condition on the weight *d* _{ k } = *k* ^{−1} exp(ln^{ β } *k*), 0 ≤ β < 1. And in a sense, our results have reached the optimal form.

### Keywords

almost sure central limit theorem weight product of sums of partial sums### MR Subject Classification

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© Editorial Committee of Applied Mathematics-A Journal of Chinese Universities and Springer-Verlag Berlin Heidelberg 2012