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

TL;DR

This paper establishes a central limit theorem for an estimator of the influence probability in a large system of interacting Hawkes processes, under subcritical conditions, with partial observations of the system.

Contribution

It provides the first CLT for the influence parameter estimator in a partially observed, mean-field Hawkes process system in the subcritical regime.

Findings

Proves a CLT for the influence parameter estimator.

Shows the estimator's asymptotic normality under subcritical conditions.

Analyzes the impact of partial observation on estimation accuracy.

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 $\mu > 0$ and, additionally, at an {\it excitation} rate of the form $N^{-1} \sum_{j=1}^{N} \theta_{ij} \int_{0}^{t} \phi(t-s)\,dZ_s^{j,N}$, 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 $\phi \colon \mathbb{R}{+} \to \mathbb{R}{+}$. 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 $\Lambda p < 1$.