Sample Path Large Deviations for Multivariate Heavy-Tailed Hawkes Processes and Related Lévy Processes
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908343613784064 |
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| author | Blanchet, Jose Laeven, Roger J. A. Wang, Xingyu Zwart, Bert |
| author_facet | Blanchet, Jose Laeven, Roger J. A. Wang, Xingyu Zwart, Bert |
| contents | In this paper, we develop sample path large deviations for multivariate Hawkes processes with heavy-tailed mutual excitation rates. Our results address a broad class of rare events in Hawkes processes at the sample path level and, via the cluster representation of Hawkes processes and a recent result on the tail asymptotics of the cluster sizes, unravel the most likely configurations of (multiple) large clusters that could trigger the target events. Our proof hinges on establishing the asymptotic equivalence, in terms of M-convergence, between a suitably scaled multivariate Hawkes process and a coupled Lévy process with multivariate hidden regular variation. Hence, along the way, we derive a sample path large deviations principle for a class of Lévy processes with multivariate hidden regular variation, which not only plays an auxiliary role in our analysis but is also of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01119 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sample Path Large Deviations for Multivariate Heavy-Tailed Hawkes Processes and Related Lévy Processes Blanchet, Jose Laeven, Roger J. A. Wang, Xingyu Zwart, Bert Probability In this paper, we develop sample path large deviations for multivariate Hawkes processes with heavy-tailed mutual excitation rates. Our results address a broad class of rare events in Hawkes processes at the sample path level and, via the cluster representation of Hawkes processes and a recent result on the tail asymptotics of the cluster sizes, unravel the most likely configurations of (multiple) large clusters that could trigger the target events. Our proof hinges on establishing the asymptotic equivalence, in terms of M-convergence, between a suitably scaled multivariate Hawkes process and a coupled Lévy process with multivariate hidden regular variation. Hence, along the way, we derive a sample path large deviations principle for a class of Lévy processes with multivariate hidden regular variation, which not only plays an auxiliary role in our analysis but is also of independent interest. |
| title | Sample Path Large Deviations for Multivariate Heavy-Tailed Hawkes Processes and Related Lévy Processes |
| topic | Probability |
| url | https://arxiv.org/abs/2504.01119 |