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Is there a nice limiting distribution of $\max( X_1,X_2,...,X_n) $ as $n$ goes to $\infty$, assuming that they are iid normal distributions with variance $\sigma^2$.

This is almost certainly a well known problem with a clever proof and nice solution, but I've been digging around and haven't found anything.

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    $\begingroup$ Rick Durrett's probability text has this as a fun problem. In the third edition, it's on page 83. $\endgroup$ – cardinal Aug 16 '12 at 10:02
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With $M_n:= \mathrm{max}(X_1,\,X_2,\,\dots,\,X_n)$ it can be shown that $(M_n-b_n)/a_n$ is approximately Gumbel for some known $a_n>0$ and $b_n$. See http://www.panix.com/~kts/Thesis/extreme/extreme2.html and the herein quoted "example 1.1.7" from the book by de Haan and Ferreira: Extreme Value theory, an Introduction.

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    $\begingroup$ +1 Great answer and a good book recommendation. There are several other good books on extreme value theory including the classic by Gumbel and the books by Galambos and the one book Leadbetter, Lindgren and Rootzen on the extension to stationary stochastic processes. A new and very readable recent book is the one by Stuart Coles. It is worth mentioning that the cumulative cdf for the Gumbel distribution exp(-e$^-$$^x$). $\endgroup$ – Michael R. Chernick Aug 16 '12 at 10:37
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Check the book Tail Risk of Hedge Funds: An Extreme Value Application, chapter 3, section 3.1. They mention that the limiting distribution of the maxima follows either Gumbel, Frechet or Weibull distribution, whatever the parent distribution F.

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