One consequence of the Berry-Esseen approach to the Central Limit theorem, states the following. Let $X\_1,\ldots ,X\_n$ be real valued random variables, all independent, with ${\mathbb E}(X\_i) = 0$, ${\mathbb E}(X\_i^2) = \sigma \_i^2$ 
which satisfies $|X\_i| \leq c$ for all $i\in [n]$ and $\sum \_{i\in [n]} \sigma \_i^2 = 1$. Set $S\_n = \sum \_{i\in [n]} X\_i$. Then for all $t \in {\mathbb R}$ we have 
\begin{equation\*}
 {\mathbb P}(S\_n \leq t) = {\mathbb P}({\cal Z} \leq t) + O(c),
\end{equation\*}
where ${\cal Z}$ is a standard normal random variable. 
My question is simply whether this bound can be improved if we additionally know that ${\mathbb E}(X\_i^3) = 0$ for all $i\in [n]$? Intuitively, it seems that this may be improved to
\begin{equation\*}
 {\mathbb P}(S\_n \leq t) = {\mathbb P}({\cal Z} \leq t) + O(c^2).
\end{equation\*} 
Any help / references are appreciated. Thanks!



Improved bounds in the Berry-Esseen theorem
Consider $X\_i$ which are equally likely to be $n^{-1/2}$ or $-n^{-1/2}$. 


These variables are essentially as nice as you could possibly want in terms of moments: The odd moments are all equal to $0$, and $E(|X\_i|^k)$ is as small as possible given the variance. 


However, the usual Berry-Esseen inequality is sharp up to constants for them ($S\_n$ has jumps corresponding to single values taken on with probability proportional to $n^{-1/2}$, while $\mathcal{Z}$ is continuous). 


This suggests that, in general, small moments are not in and of themselves enough to improve Berry Esseen. 


