Central limit theorem for a partially observed interacting system of Hawkes processes I: subcritical case

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 $\mu >0$ and, additionally, at an {\it excitation} rate of the form $N^{-1}{\sum }_{j=1}^N{\theta }_{ij}{\int }_0^t\phi (t-s)d{Z}_s^{j,N}$, 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 $\phi \text{\hspace{0.17em}⁣}:\mathbb{R}+\to \mathbb{R}+$. Here, $\mu$ and $\phi$ 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 $\mathrm{\Lambda }p<1$.