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A famous fact in statistics is that for any i.i.d. random variables $X_1,\dots,X_n$ with mean $\mu$ and variance $\sigma^2$ $$\sqrt{n}\left(\frac1n \sum_iX_i - \mu\right)$$ approaches $N(0,\sigma^2)$ as $n\to\infty$.

This is true for all random variables, but what I haven't been able to find much about is the rate at which this happens. Presumably for fatter tailed random variables the rate is much slower than for something like a $\chi^2$.

My first question is whether the "central limit rate" is known for something like $\chi^2$ and then a fat-tailed one like the log-normal.

Secondly, I would like to know if for a given $n$, there exists an (easy to construct) distribution such that $\sqrt{n}\left(\frac1n \sum_iX_i - \mu\right)$ is far from $N(0,\sigma^2)$ in some sense.

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