Open access
Author
Date
2016-08Type
- Journal Article
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(), , 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 , in the main theorem of the paper we construct an INAR()-based family of point processes and prove its convergence to . This connection between INAR and Hawkes models will be relevant in applications. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000118013Publication status
publishedExternal links
Journal / series
Stochastic Processes and their ApplicationsVolume
Pages / Article No.
Publisher
ElsevierSubject
Hawkes process; Integer-valued time series; Weak convergence of point processes; Branching processOrganisational unit
03288 - Embrechts, Paul (emeritus) / Embrechts, Paul (emeritus)
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