Law of large numbers and central limit theorem for renewal Hawkes processes

TL;DR

This paper proves a law of large numbers and a central limit theorem for a univariate Hawkes process with renewal immigration, using a martingale approach, applicable to renewal processes with absolutely continuous interarrival times.

Contribution

It introduces a novel martingale-based method to establish limit theorems for renewal Hawkes processes with absolutely continuous interarrival distributions.

Findings

Established a uniform law of large numbers for renewal Hawkes processes.

Proved a central limit theorem for these processes.

Applicable to renewal processes with absolutely continuous interarrival times.

Abstract

A uniform law of large numbers and a central limit theorem are established via a martingale approach for a univariate Hawkes process with immigration given by a renewal process. The results are obtained for renewal processes with absolutely continuous interarrival distribution.