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I am trying to find the limiting form:

$$ \lim\limits_{n \to \infty}\sum\limits_{k=1}^n \frac{1}{k}\cos(U_k) $$

where $U_k$ are iid $U_n\in(0,2\pi)$. I really have no idea where to start. I know for a fact from $L_p$ martingale convergence it does indeed converge to some number, just not sure how to find the final form. Thanks!

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    It doesn't converge to a single number (but does converge a.s.). There's a limiting distribution, but I'm not sure how to find it. Just grinding the characteristic function got pretty ugly. – 2017-01-04
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    PDF $sech^2(x)/2$ has variance $\pi^2/12$ (which is what the limiting distribution should have)... no idea for proof though and seems a bit fat-tailed so maybe just a coincidence. – 2017-01-04

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