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 is an element of N, time series model to a corresponding model of infinite order: the INAR(infinity) 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(infinity) - and an MA(infinity) representation for the model. We compare Hawkes process properties with their INAR(infinity) counterparts. Given a Hawkes process N, in the main theorem of the paper we construct an INAR(infinity)-based family of point processes and prove its convergence to N. This connection between INAR and Hawkes models will be relevant in applications. (C) 2016 The Author. Published by Elsevier B.V.