Let $(X_{j})_{j \geq 1}$ be i.i.d. with the uniform distribution on $(-1,1)$. Let $Y_{n}=\frac{\sum_{j=1}^{n}X_{j}}{\sum_{j}^{n}X_{j}^{2}+\sum_{j=1}^{n}X_{j}^{3}}$. Show that $\sqrt{n}Y_{n}$ converges in distribution to $Z$, where $\mathcal{L}(Z)=N(0,3)$.
I'm not sure how to apply the Central Limit Theorem to what I have here. Any help you could give would be greatly appreciated!
Write $$\sqrt nY_n=\color{green}{\frac{\sum_{j=1}^nX_j}{\sqrt n}}\cdot \color{red}{\left(\frac{\sum_{j=1}^nX_j^2+\sum_{j=1}^nX_j^3}n\right)^{-1}}.$$ Green term converges to $N(0,1)$, while red one's limit can be computed with the law of large number.
Then we use Slutsky theorem.