Paper: arXiv 2401.11495

Authors: Ulrich Horst, Wei Xu

Abstract

We prove that the long-run behavior of Hawkes processes is fully determined by the average number and the dispersion of child events. For subcritical processes we provide FLLNs and FCLTs under minimal conditions on the kernel of the process with the precise form of the limit theorems depending strongly on the dispersion of child events. For a critical Hawkes process with weakly dispersed child events, functional central limit theorems do not hold. Instead, we prove that the rescaled intensity processes and rescaled Hawkes processes behave like CIR-processes without mean-reversion, respectively integrated CIR-processes. We provide the rate of convergence by establishing an upper bound on the Wasserstein distance between the distributions of rescaled Hawkes process and the corresponding limit process. By contrast, critical Hawkes process with heavily dispersed child events share many properties of subcritical ones. In particular, functional limit theorems hold. However, unlike subcritical processes critical ones with heavily dispersed child events display long-range dependencies.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper offers rigorous probability theory proofs (FLLNs, FCLTs, and Wasserstein bounds) for Hawkes processes across subcritical and critical regimes. While highly novel and mathematically sophisticated, it is a purely theoretical probability paper with virtually no empirical backtesting or real-market data analysis.

Research Flowchart

  flowchart TD
    A[Research Goal: Characterize Long-Run Behavior of Hawkes Processes] --> B[Input: Hawkes Process with Child Event Dispersion & Kernel Properties]
    B --> C[Methodology: Limit Theorem Analysis & Wasserstein Distance Bounds]
    C --> D{Process Regime Analysis}
    
    D -->|Subcritical| E[Outcomes: FLLNs & FCLTs Established under Minimal Conditions]
    D -->|Critical + Weak Dispersion| F[Outcomes: Convergence to CIR / Integrated CIR Processes No FCLT]
    D -->|Critical + Heavy Dispersion| G[Outcomes: FCLTs Hold with Long-Range Dependencies]