Solution
= Solution
For $X_j\sim\operatorname{Unif}[-j,j]$,
$$
\phi_{X_j}(t)=\frac1{2j}\int_{-j}^je^{itx}dx=\frac{\sin(jt)}{jt}.
$$
Independence therefore gives
$$
\boxed{\phi_{n^{-3/2}S_n}(\xi)
=\prod_{j=1}^n\frac{\sin(j\xi/n^{3/2})}{j\xi/n^{3/2}}
=\frac{n^{3n/2}}{\xi^n n!}\prod_{j=1}^n\sin\left(\frac{j\xi}{n^{3/2}}\right).}
$$