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The accuracy of the central limit theorem for a mean depends on the skewness of the random variables being summed.

Could someone explain that to me? Or could someone propose a book to read?

Thank you.

F.F.
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    Look at this answer http://stats.stackexchange.com/a/5349/2116, there is inequality there which involves third moments. The inequality says how accurate is central limit theorem and this accuracy depends on the third moment, which is related to skewness. You can find more of such inequalities in the book "Limit theorems of Probability Theory" by V. V. Petrov. – mpiktas Nov 21 '13 at 11:16
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    Or just read Wiki's article on Edgeworth series (http://en.wikipedia.org/wiki/Edgeworth_series). It is immediately clear that the error in the CLT is strongly depending on the skewness. For symmetric random variables, this error is much, much smaller (depends primarily on the kurtosis then). – Michael M Nov 21 '13 at 11:19

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