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Sample Path Large Deviations for Multivariate Heavy-Tailed Hawkes Processes and Related Lévy Processes

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

TL;DR

The paper addresses the problem of characterizing rare, pathwise events in multivariate Hawkes processes with heavy-tailed mutual excitation. It develops a sample-path LDP by linking Hawkes dynamics to a Lévy process with multivariate hidden regular variation (MRV) in its increments, and then proving a detailed LDP for Lévy paths under $ ext{MHRV}^*$, followed by a Hawkes-specific LDP via asymptotic equivalence and cluster-based large jumps. The key contributions include (i) a pathwise LDP for Lévy processes with MRV increments, (ii) a multivariate Hawkes LDP on the product $M_1$ topology, and (iii) a practical Monte Carlo framework for evaluating the limiting measures that describe rare-event configurations formed by multiple large clusters. This framework provides theoretical and computational tools for precise rare-event analysis and efficient simulation in systems with clustering and heavy-tailed contagion across multiple components.

Abstract

In this paper, we develop sample path large deviations for multivariate Hawkes processes with heavy-tailed mutual excitation rates. Our results address a broad class of rare events in Hawkes processes at the sample path level and, via the cluster representation of Hawkes processes and a recent result on the tail asymptotics of the cluster sizes, unravel the most likely configurations of (multiple) large clusters that could trigger the target events. Our proof hinges on establishing the asymptotic equivalence, in terms of M-convergence, between a suitably scaled multivariate Hawkes process and a coupled Lévy process with multivariate hidden regular variation. Hence, along the way, we derive a sample path large deviations principle for a class of Lévy processes with multivariate hidden regular variation, which not only plays an auxiliary role in our analysis but is also of independent interest.

Sample Path Large Deviations for Multivariate Heavy-Tailed Hawkes Processes and Related Lévy Processes

TL;DR

The paper addresses the problem of characterizing rare, pathwise events in multivariate Hawkes processes with heavy-tailed mutual excitation. It develops a sample-path LDP by linking Hawkes dynamics to a Lévy process with multivariate hidden regular variation (MRV) in its increments, and then proving a detailed LDP for Lévy paths under , followed by a Hawkes-specific LDP via asymptotic equivalence and cluster-based large jumps. The key contributions include (i) a pathwise LDP for Lévy processes with MRV increments, (ii) a multivariate Hawkes LDP on the product topology, and (iii) a practical Monte Carlo framework for evaluating the limiting measures that describe rare-event configurations formed by multiple large clusters. This framework provides theoretical and computational tools for precise rare-event analysis and efficient simulation in systems with clustering and heavy-tailed contagion across multiple components.

Abstract

In this paper, we develop sample path large deviations for multivariate Hawkes processes with heavy-tailed mutual excitation rates. Our results address a broad class of rare events in Hawkes processes at the sample path level and, via the cluster representation of Hawkes processes and a recent result on the tail asymptotics of the cluster sizes, unravel the most likely configurations of (multiple) large clusters that could trigger the target events. Our proof hinges on establishing the asymptotic equivalence, in terms of M-convergence, between a suitably scaled multivariate Hawkes process and a coupled Lévy process with multivariate hidden regular variation. Hence, along the way, we derive a sample path large deviations principle for a class of Lévy processes with multivariate hidden regular variation, which not only plays an auxiliary role in our analysis but is also of independent interest.

Paper Structure

This paper contains 21 sections, 25 theorems, 343 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multivariate Hidden Regular Variation
  4. Tail Asymptotics of Hawkes Process Clusters
  5. Sample Path Large Deviations
  6. Lévy Processes with $\mathcal{MHRV}$ Increments
  7. Multivariate Heavy-Tailed Hawkes Processes
  8. Proof of Theorem \ref{['theorem: LD for Hawkes']}
  9. $\mathbb M(\mathbb S\setminus\mathbb C)$-Convergence and Asymptotic Equivalence
  10. Details for Tail Asymptotics of Hawkes Process Clusters
  11. Counterexamples of Topology and Tail Behavior
  12. Proof for Large Deviations of Lévy Processes with $\mathcal{MHRV}$ Increments
  13. Proof of Theorem \ref{['theorem: LD for levy, MRV']}
  14. Proofs of Propositions \ref{['proposition: asymptotic equivalence, LD for Levy MRV']} and \ref{['proposition: weak convergence, LD for Levy MRV']}
  15. Proof for Large Deviations of Multivariate Heavy-Tailed Hawkes Processes
  16. ...and 6 more sections

Key Result

Theorem 2

Under Assumptions assumption: subcriticality--assumption: regularity condition 2, cluster size, July 2024, it holds for any $i \in [d]$ that That is, given $i \in [d]$ and $\bm j \subseteq [d]$ with $\bm j \neq \emptyset$, if a Borel measurable set $A \subseteq \mathbb{R}^d_+$ is bounded away from $\bar{\mathbb{R}}^d_\leqslant(\bm j,\epsilon)$ under some $\epsilon > 0$, then Here, $\bar{\mathbb{

Theorems & Definitions (64)

  • Definition 1: Multivariate Hawkes Processes
  • Definition 1: $\mathcal{MHRV}$
  • Theorem 2: Theorem 3.2 of blanchet2025tailasymptoticsclustersizes
  • Definition 1: Allocation
  • Remark 1
  • Theorem 2
  • Remark 2: Interpretations of Asymptotics \ref{['claim, finite index, theorem: LD for levy, MRV']}
  • Remark 3: Proof Sketch for Theorem \ref{['theorem: LD for levy, MRV']}
  • Remark 4: Relaxation of Assumptions
  • Theorem 3
  • ...and 54 more