Revisit on the convergence rate of normal extremes

Let (Xᵢ)1 ≤ i ≤ n be independent and identically distributed (i.i.d.) standard Gaussian random variables, and denote by X₍ₙ₎ = max1 ≤ i ≤ n Xᵢ the maximum order statistic. It is well-known in extreme value theory that the linearly normalized maximum Yₙ = aₙ(X₍ₙ₎ - bₙ), converges weakly to the standard Gumbel distribution $Λ$ as n → ∞, where aₙ > 0 and bₙ are appropriate scaling and centering constants. In this note, choosing aₙ=√2log n and bₙ = √2 log n - log log n + log (4π)2 √2 log n, we provide the exact order of this convergence under several distances including Berry-Esseen bound, W₁ 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 bₙ and aₙ.