I have a stuck to prove the following statement.
Let $\{X_n,n\geq 1\}$ be an i.i.d. sequence of Uniform $[0,1]$. Show that $\dfrac{4\sum_{i=1}^n iX_i - n^2}{n^{\frac{3}{2}}}$ converges in distribution as $n\to\infty$ and find the limiting distribution.
First, I try to compute the distribution of $\sum_{i=1}^n iX_i$ by using the characteristic function $\phi_{jX_j}(t) = \int_0^1 e^{itjx} \, dx=\dfrac{e^{itj}-1}{itj}$.
However, the product of these functions are not beautiful to move on.
May someone give me a hint?