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arXiv · 0908.0680

A general strong law of large numbers for additive arithmetic functions

Istvan Berkes·Michel Weber
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Abstract

Let $f(n)$ be a strongly additive complex valued arithmetic function. Under mild conditions on $f$, we prove the following weighted strong law of large numbers: if $ X,X_1,X_2,... $ is any sequence of integrable i.i.d. random variables, then $$ \lim_{N\to \infty} {\sum_{n=1}^N f(n) X_n \over\sum_{n=1}^N f(n)} \buildrel{a.s.}\over{=} \E X . $$

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Istvan Berkes, Michel Weber. 2009-08-05. A general strong law of large numbers for additive arithmetic functions. https://arxiv.org/abs/0908.0680

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