On the norming constants for normal maxima

2013/08/26 by Gasull, Armengol, Jolis, Maria, Utzet, Frederic
#60F05 #60G70 #62G32 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1308.5541

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.

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