Hawkes and INAR($\mathrm{\infty }$) 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($p$), $p\in \mathbb{N}$, time series model to a corresponding model of infinite order: the INAR($\mathrm{\infty }$) 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 an AR($\mathrm{\infty }$), an MA($\mathrm{\infty }$), and a branching-process representation for the model. We compare Hawkes process properties with their INAR($\mathrm{\infty }$) counterparts. Given a Hawkes process $N$, in the main theorem of the paper we construct an INAR($\mathrm{\infty }$)-based family of point processes and prove its convergence to $N$. This connection between INAR and Hawkes models will be relevant in applications.