Quantification of limit theorems for Hawkes processes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912428052185088 |
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| author | Coutin, Laure Massat, Benjamin Réveillac, Anthony |
| author_facet | Coutin, Laure Massat, Benjamin Réveillac, Anthony |
| contents | In this article, we fill a gap in the literature regarding quantitative functional central limit theorems (qfCLT) for Hawkes processes by providing an upper bound for the convergence of a nearly unstable Hawkes process toward a Cox-Ingersoll-Ross (CIR) process. Note that in this case no speed of convergence has been established even for one-dimensional marginals; we provide in this paper a control in terms of a supremum norm in $2$-Wasserstein distance. To do so, we make use of the so-called Poisson imbedding representation and provide a qfCLT formulation in terms of a Brownian sheet. Incidentally, we construct an optimal coupling between a rescaled bi-dimensional Poisson random measure and a Brownian sheet with respect to the $2$-Wasserstein distance and analyze the asymptotic quality of this coupling in detail. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_21273 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantification of limit theorems for Hawkes processes Coutin, Laure Massat, Benjamin Réveillac, Anthony Probability In this article, we fill a gap in the literature regarding quantitative functional central limit theorems (qfCLT) for Hawkes processes by providing an upper bound for the convergence of a nearly unstable Hawkes process toward a Cox-Ingersoll-Ross (CIR) process. Note that in this case no speed of convergence has been established even for one-dimensional marginals; we provide in this paper a control in terms of a supremum norm in $2$-Wasserstein distance. To do so, we make use of the so-called Poisson imbedding representation and provide a qfCLT formulation in terms of a Brownian sheet. Incidentally, we construct an optimal coupling between a rescaled bi-dimensional Poisson random measure and a Brownian sheet with respect to the $2$-Wasserstein distance and analyze the asymptotic quality of this coupling in detail. |
| title | Quantification of limit theorems for Hawkes processes |
| topic | Probability |
| url | https://arxiv.org/abs/2503.21273 |