On the norming constants for normal maxima

Abstract

Given n independent standard normal random variables, it is well known that their maxima Mn can be normalized such that their distribution converges to the Gumbel law. In a remarkable study, Hall proved that the Kolmogorov distance dn between the normalized Mn and its associated limit distribution is less than 3/log⁡n. In the present study, we propose a different set of norming constants that allow this upper bound to be decreased with dn≤C(m)/log⁡n for n≥m≥5. Furthermore, the function C(m) is computed explicitly, which satisfies C(m)≤1 and limm→∞⁡C(m)=1/3. As a consequence, some new and effective norming constants are provided using the asymptotic expansion of a Lambert W type function.