Let $X_1, X_2, \ldots$ be a sequence of equally distributed independent random variables with $\mathbb P(X_i = 2) = \mathbb P(X_i = -2) = 0.5$. Consider a sequence of random variables $Y_n = {(n^{-1} \sum_{i = 1}^n X_i^3)}^2 - n^{-1} \sum_{i = 1}^n X_i^6$. One asks to find $a$ and $\sigma^2$, s. t. $\sqrt n (Y_n - a) \rightarrow \mathcal N(0, \sigma^2)$ (if they exist).
I managed to show that $Y_n$ converge almost surely to $64$. Now I have no idea how to develop this problem and I would really appreciate if somebody could provide me with some hint or a reference!