I've been working on the following problem:
Let $(X_j)_{j\geq 1}$ be independent random variables having the uniform $(-j,j)$ distribution.
a) Prove that $\lim_{n\to \infty} \frac{S_n}{n^{3/2}}=Z $ in distribution, where $Z\sim N(0,\frac{1}{9})$
Where $S_n=\sum_{i=1}^n X_i$
$\textbf{Attempt at solution}$
1) The characteristic function of $X_j$ is $\phi_{X_j}(t)=\frac{sin(tj)}{tj}$
2) Then, the characteristic function of $\frac{S_n(t)}{n^{3/2}}$ is $\phi_{\frac{S_n}{n^{3/2}}}(t)=\prod_{i=1}^n\frac{sin(\frac{t}{n^{3/2}}i)}{\frac{t}{n^{3/2}}i}$.
3) If one can show that the limit as $n\to \infty$ of the above expression is $e^{-u^2/18}$ then that should finish the proof. However, I don't see an easy way to evaluate that limit.