Revisit on the convergence rate of normal extremes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913939165544448 |
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| author | Ma, Yutao Tian, Bingjie |
| author_facet | Ma, Yutao Tian, Bingjie |
| contents | 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 $Λ$ as $n \to \infty$, where $a_n > 0$ and $b_n$ are appropriate scaling and centering constants. In this note, choosing $$a_n=\sqrt{2\log n}\quad \text{and}\quad b_n = \sqrt{2 \log n} - \frac{\log \log n + \log (4π)}{2 \sqrt{2 \log n}},$$ 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 $a_n.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09496 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Revisit on the convergence rate of normal extremes Ma, Yutao Tian, Bingjie Probability 60G70, 60B20, 60B10 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 $Λ$ as $n \to \infty$, where $a_n > 0$ and $b_n$ are appropriate scaling and centering constants. In this note, choosing $$a_n=\sqrt{2\log n}\quad \text{and}\quad b_n = \sqrt{2 \log n} - \frac{\log \log n + \log (4π)}{2 \sqrt{2 \log n}},$$ 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 $a_n.$ |
| title | Revisit on the convergence rate of normal extremes |
| topic | Probability 60G70, 60B20, 60B10 |
| url | https://arxiv.org/abs/2507.09496 |