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October 19, 2025Open Access

Scaling limit for supercritical nearly unstable Hawkes processes with heavy tail

LXLiping XuHefei University of Technology AZAn ZhangChongqing University

Key Points

The study finds that the rescaled Hawkes process converges in law to an integrated fractional Cox Ingersoll Ross process.
As n approaches infinity, the asymptotic behavior mirrors that of the subcritical case but with differing coefficients.
This work contributes to understanding the heavy tail characteristics of supercritical Hawkes processes.
The findings can have significant implications for modeling event-driven systems in various fields, including finance and neuroscience.

Abstract

In this paper, we establish the asymptotic behavior of supercritical nearly unstable Hawkes processes with a power law kernel. We find that, the Hawkes process in our context admits a similar equation to that in MR3563196 for subcritical case. In particular, the rescaled Hawkes process (Zⁿ₍ₓ/n^2) ₓ₀, ₁ converges in law to a kind of integrated fractional Cox Ingersoll Ross process with different coefficients from that in MR3563196, as n tends to infinity.

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