Deviation of empirical average from the mean for the fourth moment

Question

Suppose that $\{ X_n \}_{n \in \mathbb{N}}$ is a sequence of real i.i.d. random variables (with finite fourth order moments) with mean $\mu$. We want to show that there exists a constant $C>0$ (independent of $N$) such that

$$ \mathbb{E} \bigg[ \bigg( \frac{1}{N} \sum_{i=1}^N X_i - \mu \bigg)^4 \bigg] \leq C N^{-2}. $$

By direct expansion, we get \begin{eqnarray} \mathbb{E} \bigg[ \bigg( \frac{1}{N} \sum_{i=1}^N X_i - \mu \bigg)^4 \bigg] & = & \mathbb{E} \bigg[ \bigg( \frac{1}{N} \sum_{i=1}^N X_i \bigg)^4 \bigg] - 4 \mathbb{E} \bigg[ \bigg( \frac{1}{N} \sum_{i=1}^N X_i \bigg)^3 \mu \bigg] \nonumber \\ & & +6 \mathbb{E} \bigg[ \bigg( \frac{1}{N} \sum_{i=1}^N X_i \bigg)^2 \mu^2 \bigg] - 4 \mathbb{E} \bigg[ \bigg( \frac{1}{N} \sum_{i=1}^N X_i \bigg) \mu^3 \bigg] + \mu^4. \end{eqnarray} Clearly, by independence, the first two terms converge in $N$ with the order $O(N^{-3})$ and $O(N^{-2})$ respectively. But how about the remaining terms? Any ideas?

Answer 1

First, considering $X'_i:=X_i-\mu$, it suffices to treat the case where $\mu=0$. To do so, we expand $\left(\sum_{i=1}^nX_i\right)^4$ as $\sum_{i_1,i_2,i_3,i_4=1}^nX_{i_1}X_{i_2}X_{i_3}X_{i_4}$, take the expectation and examine the following cases:

1. All the indices $i_1$, $i_2$, $i_3$ and $i_4$ are equal;
2. two are equal to some $j$, the two others to some $k\neq j$;
3. two are equal to some $j$, the two others are different values than $j$ and different between themselves;
4. three are equal to some $j$, the other is different from $j$;
5. All the indices are different.