Large time behavior of critical marked Hawkes processes with heavy tailed marks and related branching particle systems

TL;DR

This paper analyzes the long-term behavior of critical marked Hawkes processes with heavy-tailed marks, showing convergence to stable Lévy processes under certain conditions, and employs branching process techniques.

Contribution

It extends previous work by studying the case of small eta, revealing new convergence results for critical Hawkes processes with heavy tails.

Findings

Event counting process converges to a spectrally positive rac{1}{1+eta} stable Lévy process.

Convergence holds in law in the Skorokhod space with M_1 topology.

The approach uses branching representations to analyze more general processes.

Abstract

We study large time behavior of critical marked Hawkes processes and related branching particle systems. In case of marked Hawkes processes we assume that the kernel function has multiplicative form and the marks corresponding to the events are nonnegative and are assigned independently from a common distribution. This distribution is in the normal domain of attraction of a $(1+\beta)$-stable law with $0<\beta<1$. Moreover, we assume that the mean number of events triggered by a single event is equal to $1$ (criticality). We show that, as the time is speeded up, if $\beta$ is small enough then, the event counting process, appropriately normalized, converges to a spectrally positive $1/(1+\beta)$ stable L\'evy process. The convergence holds in law in the Skorokhod space of c\`adl\`ag functions equipped with $M_1$ topology. We also study a borderline case. The present paper complements the results of [A.Talarczyk:``A generalized central limit theorem for critical marked Hawkes processes'', arXiv:2504.11612], where the same model was studied in case of ``large'' $\beta$. We employ techniques involving a branching representation of marked Hawkes processes. This approach allows to study more general branching processes with branching mechanism in the normal domain of attraction of $(1+\beta)$-stable law.