Central limit theorem for a partially observed interacting system of Hawkes processes I: subcritical case
Chenguang Liu, Liping Xu, An Zhang
TL;DR
This work analyzes a large network of Hawkes processes with Erdős–Rényi connectivity under partial observation to infer the edge probability $p$ in the subcritical regime $\Lambda p<1$. It introduces three estimators based on partial data and derives their limit theorems via a mean-field expansion using the random matrix $Q_N=(I-\Lambda A_N)^{-1}$, martingale techniques, and intricate decompositions. The main contribution is a complete multi-regime central limit theorem for the estimator $\hat p_{N,K,t}$, with explicit scalings and limiting variances in each regime, plus consistent plug-in estimation of nuisance parameters $\mu$ and $\Lambda$ to build confidence intervals. The results enable reliable inference of network connectivity in large-scale Hawkes systems from partial observations, with potential applications in neuroscience and social network analysis.
Abstract
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 $Λp < 1$.