# Distribution of sum of inverses with random signs

Let $$(Z_n)_{n\geq 0}$$ be a sequence of i.i.d. random variables with $$\mathbb{P}(Z_i=1)=\mathbb{P}(Z_i=-1)=1/2$$. Define $$S_n=\sum_{k=1}^nZ_k/k$$. Since $$(S_n)_{n\geq 0}$$ is a martingale that is bounded in $$L^p$$ for $$p>1$$, it converges a.s. and in $$L^p$$ to some r.v. $$S_\infty$$.

Can we determine the distribution of $$S_\infty$$?

• Well, the even moments $<S^{2j}_\infty>$ are $\zeta(2j)$, since $<S^{2j}_\infty>=\sum_k{<Z^{2j}> \over k^{2j}}$ Jan 7, 2021 at 9:05
• @user619894 We can find the characteristic function which is $\prod_{n=1}^\infty\cos(\xi/n)$. It seems that $S_\infty$ has a density but I'm not sure it can be expressed in terms of usual functions Jan 7, 2021 at 12:03
• The moments are finite, doesn't that imply that the PDF exists? Jan 7, 2021 at 14:16
• I'm not sure I see which property you're referring to? One other way to do things is to say that the characteristic function is integrable so $S_\infty$ has a density. I would like to know how to write this density but I'm not able to find anything conclusive Jan 7, 2021 at 14:25
• Here is a AMM article about this distribution, preprint here (click on "random harmonic series"). Jan 8, 2021 at 23:48