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 $a_n.$
From: Yutao Ma [view email]
[v1]
Sun, 13 Jul 2025 05:12:32 UTC (14 KB)
[v2]
Tue, 18 Aug 2026 00:15:22 UTC (16 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。