Tail Asymptotics of Cluster Sizes in Multivariate Heavy-Tailed Hawkes Processes

Jose Blanchet; Roger J. A. Laeven; Xingyu Wang; Bert Zwart

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arXiv · 2503.01004

Tail Asymptotics of Cluster Sizes in Multivariate Heavy-Tailed Hawkes Processes

Jose Blanchet·Roger J. A. Laeven·Xingyu Wang·Bert Zwart

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Abstract

We examine a distributional fixed-point equation related to a multi-type branching process that is key in the cluster sizes analysis of multivariate heavy-tailed Hawkes processes. Specifically, we explore the tail behavior of its solution and demonstrate the emergence of a form of multivariate hidden regular variation. Large values of the cluster size vector result from one or several significant jumps. A discrete optimization problem involving any given rare event set of interest determines the exact configuration of these large jumps and the degree of hidden regular variation. Our proofs rely on a detailed probabilistic analysis of the spatiotemporal structure of multiple large jumps in multi-type branching processes.

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BibTeX RIS

Jose Blanchet, Roger J. A. Laeven, Xingyu Wang, Bert Zwart. 2025-03-02. Tail Asymptotics of Cluster Sizes in Multivariate Heavy-Tailed Hawkes Processes. https://arxiv.org/abs/2503.01004

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