Tail Asymptotics of Cluster Sizes in Multivariate Heavy-Tailed Hawkes Processes
Jose Blanchet, Roger J. A. Laeven, Xingyu Wang, Bert Zwart
TL;DR
The paper characterizes the tail behavior of multivariate Hawkes process cluster sizes by solving a sub-critical, heavy-tailed multi-type branching fixed-point equation. It introduces a novel multivariate hidden regular variation framework, $\mathcal{MHRV}$, defined on cones in $\mathbb{R}^d_+$ and connected to polar coordinates via a polar transform $\Phi$, to capture direction-dependent tail behavior beyond classical MRV. A pruned-cluster decomposition and a type-based decomposition reveal how multiple big jumps across generations jointly shape extreme cluster sizes, leading to precise $\mathcal{MHRV}^*$ tail asymptotics with rate functions $\lambda_{\bm j}(\cdot)$ and limiting measures $\mathbf C_i^{\bm I}$. The main result identifies, for a general rare event set $A$, the dominating cone $\bm j(A)$ that minimizes the tail index, via a discrete optimization that determines the most likely jump configuration contributing to large clusters. These results underpin companion work on sample-path large deviations for multivariate heavy-tailed Hawkes processes, enabling refined rare-event analysis and simulation for clustered risk.
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.