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arxiv: 2301.07585 · v2 · pith:O537OOKCnew · submitted 2023-01-18 · 🧮 math.PR

Large deviations for the mean-field limit of Hawkes processes

Fuqing Gao , Lingjiong Zhu This is my paper

Pith reviewed 2026-05-25 08:52 UTC · model grok-4.3

classification 🧮 math.PR
keywords Hawkes processeslarge deviationsmean-field limitpoint processestime-inhomogeneous Poisson process
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The pith

The mean-field limit of multidimensional nonlinear Hawkes processes satisfies a large deviation principle.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Hawkes processes have intensities that depend on their past events in a non-Markovian way. When many such processes interact in high dimension, their collective behavior converges to a single time-inhomogeneous Poisson process whose intensity evolves according to the nonlinear feedback rule. The paper proves that this limiting Poisson process obeys a large deviation principle, supplying the exponential rate at which the probability of atypical paths decays with system size. A reader cares because the principle supplies sharp quantitative control on the chance of rare collective behaviors in large interacting point-process systems.

Core claim

The paper's central result is a large deviation principle for the mean-field limit of the multidimensional nonlinear Hawkes process, which converges to a time-inhomogeneous Poisson process.

What carries the argument

The large deviation principle for the empirical measure of the interacting Hawkes processes in the mean-field scaling limit.

If this is right

  • The rate function gives the leading exponential cost of any deviation from the typical mean-field trajectory.
  • Rare-event probabilities in finite but large systems can be approximated by the large-deviation rate.
  • The result supplies a tool for analyzing the most likely paths taken by atypical collective behavior.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same scaling limit and rate function could be used to study stability thresholds in high-dimensional systems modeled by Hawkes processes.
  • A moderate-deviations principle or central-limit theorem for fluctuations around the limit would be a natural next quantitative statement.
  • The framework may apply directly to other mean-field point-process models whose interaction is history-dependent.

Load-bearing premise

The multidimensional nonlinear Hawkes process admits a mean-field limit that is a time-inhomogeneous Poisson process.

What would settle it

Numerical simulation of a large but finite system in which the log-probability of a chosen atypical trajectory fails to scale linearly with dimension at the rate predicted by the candidate rate function.

read the original abstract

Hawkes processes are a class of simple point processes whose intensity depends on the past history, and is in general non-Markovian. Limit theorems for Hawkes processes in various asymptotic regimes have been studied in the literature. In this paper, we study a multidimensional nonlinear Hawkes process in the asymptotic regime when the dimension goes to infinity, whose mean-field limit is a time-inhomogeneous Poisson process, and our main result is a large deviation principle for the mean-field limit.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper studies a multidimensional nonlinear Hawkes process in the high-dimensional regime (dimension n → ∞). It identifies the mean-field limit as a time-inhomogeneous Poisson process and proves a large deviation principle (LDP) for this limiting object.

Significance. If the LDP holds under the stated conditions, the result supplies a new large-deviation statement for the empirical behavior of interacting point processes in the mean-field limit. This is of interest for rare-event analysis in high-dimensional self-exciting systems; the explicit identification of the limit as an inhomogeneous Poisson process is a concrete strength.

minor comments (2)
  1. The abstract states the main result but the introduction should include a brief comparison with existing LDPs for Hawkes processes (e.g., those obtained via the Poissonization or via the linear case) to clarify the novelty of the nonlinear multidimensional setting.
  2. Notation for the intensity function and the interaction kernel should be made uniform between the finite-n process and the limiting Poisson process; currently the same symbols appear to be reused without explicit redefinition in the limit section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work on large deviations for the mean-field limit of Hawkes processes. The recommendation of minor revision is appreciated. No major comments were provided in the report, so we have no specific points to address point-by-point.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained theorem

full rationale

The paper states its main result as a large deviation principle (LDP) for the mean-field limit of a multidimensional nonlinear Hawkes process, with the limit identified as a time-inhomogeneous Poisson process. No equations, parameter fits, self-citations, or ansatzes are visible in the provided abstract or setup that would reduce the LDP claim to a tautology or input by construction. The result is presented as a theorem under stated assumptions, with no load-bearing steps that equate predictions to fitted quantities or import uniqueness via self-citation chains. This is the standard case of an independent mathematical derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard existence and uniqueness assumptions for solutions of Hawkes processes and on the validity of the mean-field convergence to a Poisson process; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Existence of unique solutions to the multidimensional nonlinear Hawkes process under suitable conditions on the intensity function and interaction kernel
    Required for the mean-field limit to be well-defined as stated.

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discussion (0)

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Reference graph

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