On the norming constants for normal maxima

Armengol Gasull; Maria Jolis; Frederic Utzet

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arXiv · 1308.5541

On the norming constants for normal maxima

Armengol Gasull·Maria Jolis·Frederic Utzet

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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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Armengol Gasull, Maria Jolis, Frederic Utzet. 2013-08-26. On the norming constants for normal maxima. https://arxiv.org/abs/1308.5541

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