Let $\{X_n\}_{n=1}^\infty$ be an i.i.d. sequence of random variables that are distributed uniformly on $\{\pm 1\}$.
Define $$Y=\sum_{n=1}^\infty \frac{X_n}{n}$$ Can anything meaningful be said about the distribution of $Y$? I know that the sum converges almost always (w.p. $1$) and that the distribution will be symmetric around $0$, but I couldn't find anything else.