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The Berry-Esseen theorem is a quantitative version of the central limit theorem which sets an upper bound on deviation from normality based on the sample size $n$. The equation for identically distributed random variables is $sup|F_n(x)-\phi(x)| \le C_0*\frac{\rho}{\sigma^3*\sqrt{n}}$ where $F_n$ is the centered normalized cdf of your sample and $\phi$ is the standard normal cdf. Or at least, that's my understand of it.

I'm confused how this equation makes sense in the context of symmetric equations. When the skewness $\rho$ is zero the right side of the equation is zero. Since it should be an upper bound, that implies that the supremum of the deviation from normality (left side of the equation) is zero and not dependent on the sample size $n$.

Help? Am I misunderstanding the purpose of the equation?

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up vote 4 down vote accepted

The $\rho$ in the Berry-Essen theorem is defined as $\rho =\mathbb{E}\left(|X|^3\right)$. It is explicitly positive assuming $\sigma>0$. In other words, you may be conflating $\rho$ as the correlation coefficient, which it is not.

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