Renewal Time Points for Hawkes Processes

TL;DR

This paper introduces a renewal framework for Hawkes processes, enabling their analysis as Markov chains and deriving asymptotic results like CLTs, with applications to neural network modeling.

Contribution

It develops a renewal approach for nonlinear and age-dependent Hawkes processes, facilitating their analysis as Markov chains and deriving new asymptotic properties.

Findings

Constructed renewal times for Hawkes processes

Established moment results for renewal times

Proved functional and time-average CLTs

Abstract

In the last decade Hawkes processes have received much attention as models for functional connectivity in neural spiking networks and other dynamical systems with a cascade behavior. In this paper we establish a renewal approach for analyzing this process. We consider the ordinary nonlinear Hawkes process as well as the more recently described age dependent Hawkes process. We construct renewal-times and establish moment results for these. This gives rise to study the Hawkes process as a Markov chain. As an application, we prove asymptotic results such as a functional CLT and a time-average CLT.