Let .Xi /i1 be independent standard normal random variables and

Show that the sequence anMn a2n converges in distribution to the probability measure Q on R with distribution function FQ.c/ D exp.e c/ ; c 2 R:
Q is known as the doubly exponential distribution or, after Emil J. Gumbel (1891–1966), the (standard) Gumbel distribution. It is one of the so-called extreme value distributions that appear as the asymptotic distribution of rescaled maxima.

Mn max (XX), an= 2logn log(logn) - log(4).