Central limit theorem for a partially observed interacting system of Hawkes processes

TL;DR

This paper establishes a central limit theorem for an estimator of the connection probability in a large, partially observed system of interacting Hawkes processes, accounting for unknown parameters and different critical regimes.

Contribution

It introduces a CLT for the estimator of the interaction probability in a partially observed Hawkes system, covering subcritical and supercritical cases.

Findings

CLT for the estimator of p in large systems

Results valid for both subcritical and supercritical regimes

Handles unknown nuisance parameters μ and φ

Abstract

We observe the actions of a $K$ sub-sample of $N$ individuals up to time $t$ for some large $K\le N$. We model the relationships of individuals by i.i.d. Bernoulli($p$)-random variables, where $p\in (0,1]$ is an unknown parameter. The rate of action of each individual depends on some unknown parameter $\mu> 0$ and on the sum of some function $\phi$ of the ages of the actions of the individuals which influence him. The parameters $\mu$ and $\phi$ are considered as nuisance parameters. The aim of this paper is to obtain a central limit theorem for the estimator of $p$ that we introduced in \cite{D}, both in the subcritical and supercritical cases.