Central limit theorem for a partially observed interacting system of Hawkes processes I: subcritical case
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arXiv
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2026
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| author | Liu, Chenguang Xu, Liping Zhang, An |
| author_facet | Liu, Chenguang Xu, Liping Zhang, An |
| contents | We consider a system of $N$ Hawkes processes and observe the actions of a subpopulation of size $K \le N$ up to time $t$, where $K$ is large. The influence relationships between each pair of individuals are modeled by i.i.d.Bernoulli($p$) random variables, where $p \in [0,1]$ is an unknown parameter. Each individual acts at a {\it baseline} rate $μ> 0$ and, additionally, at an {\it excitation} rate of the form $N^{-1} \sum_{j=1}^{N} θ_{ij} \int_{0}^{t} ϕ(t-s)\,dZ_s^{j,N}$, which depends on the past actions of all individuals that influence it, scaled by $N^{-1}$ (i.e. the mean-field type), with the influence of older actions discounted through a memory kernel $ϕ\colon \mathbb{R}{+} \to \mathbb{R}{+}$. Here, $μ$ and $ϕ$ are treated as nuisance parameters. The aim of this paper is to establish a central limit theorem for the estimator of $p$ proposed in \cite{D}, under the subcritical condition $Λp < 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01189 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Central limit theorem for a partially observed interacting system of Hawkes processes I: subcritical case Liu, Chenguang Xu, Liping Zhang, An Probability Statistics Theory Mathematical Finance We consider a system of $N$ Hawkes processes and observe the actions of a subpopulation of size $K \le N$ up to time $t$, where $K$ is large. The influence relationships between each pair of individuals are modeled by i.i.d.Bernoulli($p$) random variables, where $p \in [0,1]$ is an unknown parameter. Each individual acts at a {\it baseline} rate $μ> 0$ and, additionally, at an {\it excitation} rate of the form $N^{-1} \sum_{j=1}^{N} θ_{ij} \int_{0}^{t} ϕ(t-s)\,dZ_s^{j,N}$, which depends on the past actions of all individuals that influence it, scaled by $N^{-1}$ (i.e. the mean-field type), with the influence of older actions discounted through a memory kernel $ϕ\colon \mathbb{R}{+} \to \mathbb{R}{+}$. Here, $μ$ and $ϕ$ are treated as nuisance parameters. The aim of this paper is to establish a central limit theorem for the estimator of $p$ proposed in \cite{D}, under the subcritical condition $Λp < 1$. |
| title | Central limit theorem for a partially observed interacting system of Hawkes processes I: subcritical case |
| topic | Probability Statistics Theory Mathematical Finance |
| url | https://arxiv.org/abs/2601.01189 |