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