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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.21155 |
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| _version_ | 1866910240756203520 |
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| author | Zhang, Chunxu Miao, Baiqi Mao, Tiantian |
| author_facet | Zhang, Chunxu Miao, Baiqi Mao, Tiantian |
| contents | We study asymptotic probabilities of attaining the maximum in heterogeneous Gaussian samples. In the two-group setting, the first sample has variance $1$ and size $n_1$, while the second has variance $σ^2>1$ and size $n_2$. We investigate the probability that the maximum of the standard-variance group exceeds that of the high-variance group. Using the classical extreme-value normalization for Gaussian maxima together with a second-order comparison of the centering terms, we show that this probability admits a non-degenerate limit if and only if $n_1\sim C n_2^{σ^2}(\log n_2)^{-(σ^2-1)/2}$ as $n_1,n_2\to\infty$ for some $C\in(0,\infty)$. In that regime, the limit admits an integral representation. Outside the critical regime, the comparison necessarily degenerates to $0$ or $1$. We then extend the analysis to finitely many independent Gaussian groups and obtain a generalized integral representation for the limiting winning probabilities. The results provide a complete asymptotic classification for this maximum-comparison problem |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21155 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic Probabilities of Attaining the Maximum in Heterogeneous Gaussian Samples Zhang, Chunxu Miao, Baiqi Mao, Tiantian Probability We study asymptotic probabilities of attaining the maximum in heterogeneous Gaussian samples. In the two-group setting, the first sample has variance $1$ and size $n_1$, while the second has variance $σ^2>1$ and size $n_2$. We investigate the probability that the maximum of the standard-variance group exceeds that of the high-variance group. Using the classical extreme-value normalization for Gaussian maxima together with a second-order comparison of the centering terms, we show that this probability admits a non-degenerate limit if and only if $n_1\sim C n_2^{σ^2}(\log n_2)^{-(σ^2-1)/2}$ as $n_1,n_2\to\infty$ for some $C\in(0,\infty)$. In that regime, the limit admits an integral representation. Outside the critical regime, the comparison necessarily degenerates to $0$ or $1$. We then extend the analysis to finitely many independent Gaussian groups and obtain a generalized integral representation for the limiting winning probabilities. The results provide a complete asymptotic classification for this maximum-comparison problem |
| title | Asymptotic Probabilities of Attaining the Maximum in Heterogeneous Gaussian Samples |
| topic | Probability |
| url | https://arxiv.org/abs/2605.21155 |