A simple question on CLT in possible connection with Berry-Esseen thm I am curious about the contents while I read a note on machine learning. It could be obvious. So, please let me know if I am missing some fundamental things.
$X_1,X_2,...,X_n$ are from an i.i.d. distribution with mean $\mu$ and variance $\sigma<\infty$. Then,
by C.L.T, $s_n$ is approximately normal with mean $\mu$ and $\frac{\sigma}{\sqrt{n}}$, so that it has the density: $f_{s_n}(x)=\frac{1}{\sqrt{\frac{2\pi \sigma^2}{n}}}e^{-\frac{n}{2}\frac{(x-u)^2}{\sigma^2}}$. 
What I cannot grasp is the following comment: "This approximation is only valid for x within about $\frac{\sigma}{\sqrt{n}}$ of $\mu$". (FYI, this note is about large deviation.)
I also wondering how this statement is closely related to the Berry-Esseen thm. 
 A: There are several mistakes with the statement you give. I think a better resource for anyone interested in Machine Learning these days would be Tibshirani Hastie "Elements of Statistical Learning."
The correct statement of the CLT is that, given $X_1, X_2, \ldots, X_n \sim iid (\mu, \sigma^2)$,
$$\sqrt{n} \left(\bar{X} - \mu \right) \rightarrow_d \mathcal{N}(0, \sigma^2) $$
(note that several of these "assumptions" can be relaxed, ref Lindeberg Feller CLT)
A consequence is that the distribution of the sample mean can be approximated using a $\mathcal{N}(\bar{X}, \sigma^2/n)$ distribution. The cumulative sum $S_n$ is given by $n \bar{X}$ which would have an approximating distribution of $\mathcal{N}(n\mu, n\sigma^2)$ (recall $\mbox{Var}(aX) = a^2 \mbox{Var}(X)$).
The last problem with your statement is that you confuse the probability density function for a normal curve with a cumulative DF. The $Pr(S_n <x)$ would be best approximated with $Pr(S_n < x) \approx \Phi(\frac{x - n\mu}{\sqrt{n \sigma^2}})$ (not $\phi$)... but that seems like a silly thing to approximate when the sample mean is so much more regularly behaved... i.e. it actually converges to something.
The range in which you refer to the accuracy of the approximation is complete bunk. The CLT is certainly based on a Taylor series expansion... but making a rule of thumb about it is useless. More $n$ implies better approximating distribution, that's all we can say about it.
From what I gather, Barry-Essen has nothing to do with any of this.
All in all, please disregard the note.
