This is from and exercise of probability theory:
Let $(X_j)_{j\geq 1}$ be independent and let $X_j$ have the uniform distribution on $(-j,j)$. Show that $ \lim_{n\to\infty}\frac{S_n}{n^{3/2}}=Z\sim N(0,\frac{1}{9}) $ in distribution.
In terms of characteristic functions, it suffices to show that $ \lim_{n\to\infty}\prod_{j=1}^n\frac{\sin(jn^{-3/2}u)}{jn^{-3/2}u}=e^{-u^2/18}. $ Take $\log$ on both side one gets: $ \lim_{n\to\infty}\sum_{j=1}^n\log\frac{\sin(jn^{-3/2}u)}{jn^{-3/2}u}=-\frac{u^2}{18}. $ How can this limit be proved?