Let $\{X_n\}_{n\geq 1}$ be a sequence of i.i.d. random variables with mean zero and finite fourth moment. Let $S_n=X_1+X_2+\dots+X_n$.
Show that $$\mathbb{E}[S_{_n}^4]=\mathcal{O}(n^2)$$
I feel like this is more of a calculus problem but I don't know how to show this (simple) fact.
Rather than being a "calculus problem", it actually ends up being a bit of combinatorics.
When you apply the multinomial theorem to expand $S_n^4$, the only monomials that will result in a non-zero expectation must have each distinct variable appearing with even degree. This leaves either $X_i^4$ or $X_i^2X_j^2$. There are $O(n)$ of the first kind and $O(n^2)$ of the second kind.