Hawkes and INAR($\infty$) processes
By Matthias Kirchner. 2015
53 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Molecular Biology
View Paper: https://www.semanticscholar.org/paper/e00e999dca2789b65e98da81bbe27ac0f5628436
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In Applied Mathematics
In Molecular Biology
Citations in the corpus
- Hawkes processes and their applications to finance: a review (2018, background): “Kirchner (2016) points out the close relation of a self-exciting Hawkes process to an infinite order Integervalued Auto-Regression, INAR (∞), noting in particular the equivalent branching process constructions in the two processes.”