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.
Best Answer
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.