Learning the exogenous rate but not the distance to criticality in nearly unstable heavy-tailed Hawkes processes
Abstract
In the nearly unstable heavy-tailed regime of Jaisson and Rosenbaum, where the intensity of a linear Hawkes process converges to a rough square-root Volterra process, we ask what a record of length T reveals about the relative distance to criticality 1−ρ and the exogenous rate μ. For the first, the information is governed by ℐT=Tμ(1−ρ), not by the number of events; ℐT stays bounded although the event count diverges like T2γ, so no estimator of the relative distance to criticality is uniformly locally consistent. For the second, the Fisher information diverges because the rough limit has an atom at zero: the asymptotic information geometry is singular in the exogenous direction and, unlike the light-tailed case, has no Feller-type threshold. We prove that the joint maximum likelihood estimator of (μ,ρ), whose likelihood is jointly concave, recovers μ consistently at the random rate {QT/loglog QT}1/2 given by the observed information QT, while its estimate of the relative margin is only tight: the exogenous rate is learned although the endogenous parameter it is coupled to is not. For empty-start records QT diverges because the rough limit started at zero spends positive time at zero; on windows observed long after the start, still with the branching ratio unknown, learnability holds with probability tending to one as the window grows, through the ergodicity of the rough limit and the atom of its stationary law. We also prove finite-dimensional convergence of the stationary intensity to the stationary rough Volterra process.
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BibTeX
@misc{herreramarin2026learning,
title = {{Learning the exogenous rate but not the distance to criticality in nearly unstable heavy-tailed Hawkes processes}},
author = {Mauricio Herrera-Marín},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38268},
eprint = {2609.38268},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38268},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Learning the exogenous rate but not the distance to criticality in nearly unstable heavy-tailed Hawkes processes
AU - Mauricio Herrera-Marín
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38268
UR - https://arxiv.org/abs/2609.38268
AB - In the nearly unstable heavy-tailed regime of Jaisson and Rosenbaum, where the intensity of a linear Hawkes process converges to a rough square-root Volterra process, we ask what a record of length T reveals about the relative distance to criticality 1−ρ and the exogenous rate μ. For the first, the information is governed by ℐ_T=Tμ(1−ρ), not by the number of events; ℐ_T stays bounded although the event count diverges like T^2γ, so no estimator of the relative distance to criticality is uniformly locally consistent. For the second, the Fisher information diverges because the rough limit has an atom at zero: the asymptotic information geometry is singular in the exogenous direction and, unlike the light-tailed case, has no Feller-type threshold. We prove that the joint maximum likelihood estimator of (μ,ρ), whose likelihood is jointly concave, recovers μ consistently at the random rate {Q_T/loglog Q_T}^1/2 given by the observed information Q_T, while its estimate of the relative margin is only tight: the exogenous rate is learned although the endogenous parameter it is coupled to is not. For empty-start records Q_T diverges because the rough limit started at zero spends positive time at zero; on windows observed long after the start, still with the branching ratio unknown, learnability holds with probability tending to one as the window grows, through the ergodicity of the rough limit and the atom of its stationary law. We also prove finite-dimensional convergence of the stationary intensity to the stationary rough Volterra process.
ER -