 Functional limit theorems for nonstationary marked Hawkes processes in the high intensity regimeBo Li and Guodong Pang Stochastic Processes and their Applications, 2022, vol. 143, issue C, 285-339 Abstract: We study marked Hawkes processes with an intensity process which has a non-stationary baseline intensity, a general self-exciting function of event “ages” at each time and marks. The marks are assumed to be conditionally independent given the event times, while the distribution of each mark depends on the event time, that is, time-varying. We first observe an immigration–birth (branching) representation of such a non-stationary marked Hawkes process, and then derive an equivalent representation of the process using the associated conditional inhomogeneous Poisson processes with stochastic intensities. We consider such a Hawkes process in the high intensity regime, where the baseline intensity gets large, while the self-exciting function and distributions of the marks are unscaled, and there is no time-scaling in the scaled Hawkes process. We prove functional law of large numbers and functional central limit theorems (FCLTs) for the scaled Hawkes processes in this asymptotic regime. The limits in the FCLTs are characterized by continuous Gaussian processes with covariance structures expressed with convolution functionals resulting from the branching representation. We also consider the special cases of multiplicative self-exciting functions, and an indicator type of non-decomposable self-exciting functions (including the cases of “ceasing” and “delayed” reproductions as well as their extensions with varying reproduction rates), and study the properties of the limiting Gaussian processes in these special cases. Keywords: Marked Hawkes process; Non-stationarity; High intensity asymptotic regime; Immigration–birth representation; Functional limit theorems; Continuous Gaussian processes (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:143:y:2022:i:c:p:285-339 Ordering information: This journal article can be ordered from DOI: 10.1016/j.spa.2021.10.008 Access Statistics for this article Stochastic Processes and their Applications is currently edited by T. Mikosch More articles in Stochastic Processes and their Applications from Elsevier |
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