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I would like to understand the relationship between the asymptotic moments of order statistics and the moments of the distribution of the mother distribution. I will appreciate any references on this matter.

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Asymptotic for which limit? –  Did Jul 2 '11 at 9:20
    
as the number of samples goes to infinity –  user12847 Jul 2 '11 at 9:39
    
Then the answer very much depends on whether (what you call) the mother distribution is bounded or not, and on the rank of the order statistics you consider, either at the top, or at the bottom, or somewhere in the middle of the sample. For the latter, you might wish to consult the formula given here: en.wikipedia.org/wiki/Order_statistic#Large_sample_sizes –  Did Jul 2 '11 at 9:47
    
Yes, and what is the relation of this asymptotic variance and the variance of the mother distribution? –  user12847 Jul 2 '11 at 10:17
    
Bis repetita: the asymptotic variance of what? $X_{(1)}$? $X_{(n)}$? $X_{(k)}$ for $k=pn+o(n)$ and $p$ fixed in $(0,1)$? These all lead to very different answers. For the latter, the answer is on the WP page I linked to. –  Did Jul 2 '11 at 11:00

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