On the rate of convergence of normal extremes

Let Y n denote the largest of n independent N (0, 1) variables. It is shown that if the constants a n and b n are chosen in an optimal way then the rate of convergence of ( Y n – b n )/ a n to the extreme value distribution exp(– e –x ), as measured by the supremum metric or the Lévy metric, is proportional to 1/log n.