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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.17966 |
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| _version_ | 1866917798703267840 |
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| author | Bai, Shuyang Kulik, Rafał Wang, Yizao |
| author_facet | Bai, Shuyang Kulik, Rafał Wang, Yizao |
| contents | The stable-regenerative multiple-stable model has been shown recently to have distinct candidate extremal index and extremal index. To understand further this rare phenomenon, two more results are established here for the double-stable model. The first is the convergence of point processes for the clusters of extremes, enhancing the previous result on the weak convergence of random sup-measures. Most interestingly, the second result reveals a new phase transition at the mesoscopic level when computing the asymptotic exceedance probability over a block, $\mathbb P(\max_{k=1,\dots,d_n} X_k>b_n)$, as $n\to\infty$. Here, the mesoscopic level is referred to the fact that the block size $d_n$ is allowed to grow at the rate $n^ρ$ with $ρ\in[0,1]$, while the threshold $b_n$ is such that $\mathbb P(X_1>b_n)\sim 1/n$. The recently discovered discrepancy between the candidate extremal index and the extremal index is shown to be just a reflection of this phase transition that is prohibited by the anticlustering condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17966 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A remarkable example on clustering of extremes for regularly-varying stochastic processes Bai, Shuyang Kulik, Rafał Wang, Yizao Probability The stable-regenerative multiple-stable model has been shown recently to have distinct candidate extremal index and extremal index. To understand further this rare phenomenon, two more results are established here for the double-stable model. The first is the convergence of point processes for the clusters of extremes, enhancing the previous result on the weak convergence of random sup-measures. Most interestingly, the second result reveals a new phase transition at the mesoscopic level when computing the asymptotic exceedance probability over a block, $\mathbb P(\max_{k=1,\dots,d_n} X_k>b_n)$, as $n\to\infty$. Here, the mesoscopic level is referred to the fact that the block size $d_n$ is allowed to grow at the rate $n^ρ$ with $ρ\in[0,1]$, while the threshold $b_n$ is such that $\mathbb P(X_1>b_n)\sim 1/n$. The recently discovered discrepancy between the candidate extremal index and the extremal index is shown to be just a reflection of this phase transition that is prohibited by the anticlustering condition. |
| title | A remarkable example on clustering of extremes for regularly-varying stochastic processes |
| topic | Probability |
| url | https://arxiv.org/abs/2409.17966 |