Hawkes and INAR(∞) processes

Matthias Kirchner

Stochastic Processes and their Applications, 2016, vol. 126, issue 8, 2494-2525

Abstract: 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.

Keywords: Hawkes process; Integer-valued time series; Weak convergence of point processes; Branching process (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (6)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:126:y:2016:i:8:p:2494-2525

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DOI: 10.1016/j.spa.2016.02.008

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