Renewal Time Points for Hawkes Processes

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.

Mean field limits for interacting Hawkes processes in a diffusive regime

“…The function f N i is called the jump rate function of Z N,i . Since the founding works of [18] and [19], many probabilistic properties of Hawkes processes have been well-understood, such as ergodicity, stationarity and long time behaviour (see [5], [9], [8], [34] and [16]). A number of authors studied the statistical inference for Hawkes processes ( [30] and [36]).…”