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 $S_\infty$?