Large deviations for the maximum and reversed order statistics of Weibull-like variables
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916262365364224 |
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| author | Jansen, Sabine |
| author_facet | Jansen, Sabine |
| contents | Motivated by metastability in the zero-range process, we consider i.i.d.\ random variables with values in $\N_0$ and Weibull-like (stretched exponential) law $\mathbb P(X_i =k) = c \exp( - k^α)$, $α\in (0,1)$. We condition on large values of the sum $S_n= μn + s n^γ$ and prove large deviation principles for the rescaled maximum $M_n /n^γ$ and for the reversed order statistics. The scale is $n^γ$ with $γ= 1/(2-α)$; on that scale, the big-jump principle for heavy-tailed variables and a naive normal approximation for moderate deviations yield bounds of the same order $n^{γα} = n^{2γ-1}$, the speed of the large deviation principles. The rate function for $M_n/n^γ$ is non-convex and solves a recursive equation similar to a Bellman equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_17319 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large deviations for the maximum and reversed order statistics of Weibull-like variables Jansen, Sabine Probability 60F10, 60G50, 60K35 Motivated by metastability in the zero-range process, we consider i.i.d.\ random variables with values in $\N_0$ and Weibull-like (stretched exponential) law $\mathbb P(X_i =k) = c \exp( - k^α)$, $α\in (0,1)$. We condition on large values of the sum $S_n= μn + s n^γ$ and prove large deviation principles for the rescaled maximum $M_n /n^γ$ and for the reversed order statistics. The scale is $n^γ$ with $γ= 1/(2-α)$; on that scale, the big-jump principle for heavy-tailed variables and a naive normal approximation for moderate deviations yield bounds of the same order $n^{γα} = n^{2γ-1}$, the speed of the large deviation principles. The rate function for $M_n/n^γ$ is non-convex and solves a recursive equation similar to a Bellman equation. |
| title | Large deviations for the maximum and reversed order statistics of Weibull-like variables |
| topic | Probability 60F10, 60G50, 60K35 |
| url | https://arxiv.org/abs/2405.17319 |