TITLE: What is the limit behavior of this random sum?
QUESTION [7 upvotes]: Let $(X_n\mid n\in\mathbb{N})$ be an i.i.d. sequence of random variables taking values in $\mathbb{R}$. What can be said about the limit behavior of 
\begin{equation}
S_n:=\sum_{i=1}^n\frac{X_i}{i}
\end{equation}
as $n\rightarrow \infty$? In particular, under what conditions does $S_n$ converge (to zero?), and under what conditions does it not converge? Many thanks for any help!

REPLY [4 votes]: Let $\phi$ be the characteristic function of $X_1$. Then $\phi_n$, the characteristic function of $S_n$, is given by 
$$\phi_n(t)=\prod_{j=1}^n\phi\left(\frac tj\right),$$
so $(S_n,n\geqslant 1)$ converges in distribution if and only if for each $t\in\mathbb R$, the product $\prod_{j=1}^\infty\phi\left(\frac tj\right)$ is convergent (in the usual sense or $\phi(t/j)=0$ for some $j$).
There are case where the product is convergent, and when the product is divergent. For example, consider random variables with characteristic function $\phi(t):=e^{-|t|^\alpha}$, where $0\lt\alpha\lt 2$. Then the product is convergent if and only if $\alpha\gt 1$.
If $S_n$ converges in distribution to $0$ then $|\phi(t)|=1$ for each $t$.