A law of the iterated logarithm for arithmetic functions

Let $X,X_1,X_2,\ldots$ be a sequence of centered iid random variables. Let $f(n)$ be a strongly additive arithmetic function such that $\sum_{p < n}\tfrac{f^2(p)}{p}\to\infty$ and put $A_n= \sum_{p < n}\tfrac{f(p)}{p}$. If $\mathbf{E} X^2 <\infty$ and $f$ satisfies a Lindeberg-type condition, we prove the following law of the iterated logarithm:

$\limsup_{N\to \infty}{\sum_{n=1}^N f(n) X_n \over A_N \sqrt{2 N \log \log N}}\buildrel{a.s.}\over{=} \Vert X\Vert _2.$

We also prove the validity of the corresponding weighted strong law of large numbers in $L^1$.