Revisit on the convergence rate of normal extremes

Abstract

Let $(X_i{)}_{1\le i\le n}$ be independent and identically distributed (i.i.d.) standard Gaussian random variables, and denote by $X_{(n)}={\max }_{1\le i\le n}{X}_i$ the maximum order statistic. It is well-known in extreme value theory that the linearly normalized maximum $Y_n=a_n(X_{(n)}-b_n),$ converges weakly to the standard Gumbel distribution $\Lambda$ as $n\to \infty$, where $a_n>0$ and $b_n$ are appropriate scaling and centering constants. In this note, choosing $$a_n=\sqrt{2\log n}\text{and}{b}_n=\sqrt{2\log n}-\frac{\log \log n+\log (4\pi )}{2\sqrt{2\log n}},$$ we provide the exact order of this convergence under several distances including Berry-Esseen bound, $W_1$ distance, total variation distance, Kullback-Leibler divergence and Fisher information. We also show how the orders of these convergence are influenced by the choice of $b_n$ and $a_n.$

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