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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.16903 |
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| _version_ | 1866912142841610240 |
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| author | Baars, Justin Laeven, Roger J. A. Mandjes, Michel |
| author_facet | Baars, Justin Laeven, Roger J. A. Mandjes, Michel |
| contents | We introduce a spatiotemporal self-exciting point process $(N_t(x))$, boundedly finite both over time $[0,\infty)$ and space $\mathscr X$, with excitation structure determined by a graphon $W$ on $\mathscr{X}^2$. This graphon Hawkes process generalizes both the multivariate Hawkes process and the Hawkes process on a countable network, and despite being infinite-dimensional, it is surprisingly tractable. After proving existence, uniqueness and stability results, we show, both in the annealed and in the quenched case, that for compact, Euclidean $\mathscr X\subset\mathbb R^m$, any graphon Hawkes process can be obtained as the suitable limit of $d$-dimensional Hawkes processes $\tilde N^d$, as $d\to\infty$. Furthermore, in the stable regime, we establish an FLLN and an FCLT for our infinite-dimensional process on compact $\mathscr X\subset\mathbb R^m$, while in the unstable regime we prove divergence of $N_T(\mathscr X)/T$, as $T\to\infty$. Finally, we exploit a cluster representation to derive fixed-point equations for the Laplace functional of $N$, for which we set up a recursive approximation procedure. We apply these results to show that, starting with multivariate Hawkes processes $\tilde N^d_t$ converging to stable graphon Hawkes processes, the limits $d\to\infty$ and $t\to\infty$ commute. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_16903 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spatiotemporal Hawkes processes with a graphon-induced connectivity structure Baars, Justin Laeven, Roger J. A. Mandjes, Michel Probability 60G55, 60F15, 05C80 We introduce a spatiotemporal self-exciting point process $(N_t(x))$, boundedly finite both over time $[0,\infty)$ and space $\mathscr X$, with excitation structure determined by a graphon $W$ on $\mathscr{X}^2$. This graphon Hawkes process generalizes both the multivariate Hawkes process and the Hawkes process on a countable network, and despite being infinite-dimensional, it is surprisingly tractable. After proving existence, uniqueness and stability results, we show, both in the annealed and in the quenched case, that for compact, Euclidean $\mathscr X\subset\mathbb R^m$, any graphon Hawkes process can be obtained as the suitable limit of $d$-dimensional Hawkes processes $\tilde N^d$, as $d\to\infty$. Furthermore, in the stable regime, we establish an FLLN and an FCLT for our infinite-dimensional process on compact $\mathscr X\subset\mathbb R^m$, while in the unstable regime we prove divergence of $N_T(\mathscr X)/T$, as $T\to\infty$. Finally, we exploit a cluster representation to derive fixed-point equations for the Laplace functional of $N$, for which we set up a recursive approximation procedure. We apply these results to show that, starting with multivariate Hawkes processes $\tilde N^d_t$ converging to stable graphon Hawkes processes, the limits $d\to\infty$ and $t\to\infty$ commute. |
| title | Spatiotemporal Hawkes processes with a graphon-induced connectivity structure |
| topic | Probability 60G55, 60F15, 05C80 |
| url | https://arxiv.org/abs/2409.16903 |