Revisit on the convergence rate of normal extremes

TL;DR

This paper investigates the convergence rate of the maximum of i.i.d. Gaussian variables to the Gumbel distribution, providing precise bounds under various distances and analyzing the impact of normalization constants.

Contribution

It offers the exact order of convergence for normalized Gaussian maxima to the Gumbel law across multiple metrics, refining understanding of the convergence behavior.

Findings

Convergence rates depend on the choice of normalization constants.

Explicit bounds are derived for Berry-Esseen, Wasserstein, total variation, KL divergence, and Fisher information.

The study clarifies how normalization affects the speed of convergence to the Gumbel distribution.

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 $$a_n=\sqrt{2\log n}\quad \text{and}\quad b_n = \sqrt{2 \log n} - \frac{\log \log n + \log (4\pi)}{2 \sqrt{2 \log n}},$$ 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 $a_n.$