Hawkes and INAR(∞) processes

In this paper, we discuss integer-valued autoregressive time series (INAR), Hawkes point processes, and their interrelationship. Besides presenting structural analogies, we derive a convergence theorem. More specifically, we generalize the well-known INAR(p), p∈N, time series model to a corresponding model of infinite order: the INAR(∞) model. We establish existence, uniqueness, finiteness of moments, and give formulas for the autocovariance function as well as for the joint moment-generating function. Furthermore, we derive a branching-process-as well as an AR(∞)-and an MA(∞) representation for the model. We compare Hawkes process properties with their INAR(∞) counterparts. Given a Hawkes process N, in the main theorem of the paper we construct an INAR(∞)-based family of point processes and prove its convergence to N. This connection between INAR and Hawkes models will be relevant in applications.