On the norming constants for normal maxima

TL;DR

This paper improves the bounds on the convergence rate of the distribution of normalized maxima of standard normal variables to the Gumbel law by selecting better norming constants, leading to more precise asymptotic approximations.

Contribution

It introduces a new set of norming constants that reduce the convergence bound from 3/ log n to 1/ log n, refining previous results.

Findings

Bound on the supremum norm distance is reduced to 1/ log n

New explicit constants for normal maxima are proposed

Asymptotic expansion involves Lambert W function

Abstract

In a remarkable paper, Peter Hall [{\it On the rate of convergence of normal extremes}, J. App. Prob, {\bf 16} (1979) 433--439] proved that the supremum norm distance between the distribution function of the normalized maximum of $n$ independent standard normal random variables and the distribution function of the Gumbel law is bounded by $3/\log n$. In the present paper we prove that choosing a different set of norming constants that bound can be reduced to $1/\log n$. As a consequence, using the asymptotic expansion of a Lambert $W$ type function, we propose new explicit constants for the maxima of normal random variables.