Revisit on the convergence rate of normal extremes

Let (Xᵢ) ₁ ₈ ₍ be independent and identically distributed (i. i. d. ) standard Gaussian random variables, and denote by X (₍) = ₁ ₈ ₍ Xᵢ the maximum order statistic. It is well-known in extreme value theory that the linearly normalized maximum Yₙ = aₙ (X (₍) - 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=2 n and bₙ = 2 n - n + (4π) 2 2 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.