Paper: arXiv 2601.01189

Authors: Chenguang Liu, Liping Xu, An Zhang

Abstract

We consider a system of $N$ Hawkes processes and observe the actions of a subpopulation of size $K \le N$ up to time $t$, where $K$ is large. The influence relationships between each pair of individuals are modeled by i.i.d.Bernoulli($p$) random variables, where $p \in [0,1]$ is an unknown parameter. Each individual acts at a {\it baseline} rate $μ> 0$ and, additionally, at an {\it excitation} rate of the form $N^{-1} \sum_{j=1}^{N} θ_{ij} \int_{0}^{t} φ(t-s),dZ_s^{j,N}$, which depends on the past actions of all individuals that influence it, scaled by $N^{-1}$ (i.e. the mean-field type), with the influence of older actions discounted through a memory kernel $φ\colon \mathbb{R}{+} \to \mathbb{R}{+}$. Here, $μ$ and $φ$ are treated as nuisance parameters. The aim of this paper is to establish a central limit theorem for the estimator of $p$ proposed in \cite{D}, under the subcritical condition $Λp < 1$.

Complexity vs Empirical Score

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

Why this score: The paper is a rigorous mathematical probability study establishing a central limit theorem for parameter estimation in partially observed mean-field Hawkes processes. It features exceptionally deep measure-theoretic and stochastic calculus proofs across 56 pages, but it contains no empirical application, backtesting, or real market dataset validation.

Research Flowchart

  flowchart TD
    A[Research Goal: CLT for Network Influence Parameter p in Partially Observed Hawkes System] --> B[Model Inputs & Setup]
    
    B -->|Observed K <= N units, Baseline μ, Memory Kernel φ| C[Subcritical Regime Condition: Λp < 1]
    
    C --> D[Mean-Field Limit Analysis & Fluctuations Calculation]
    
    D --> E[Asymptotic Expansion of Estimator p_hat from Ditlevsen et al.]
    
    E --> F[Key Outcome: Central Limit Theorem for Estimator p_hat as K, N, t -> ∞]

Cited References (35)

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  8. Modeling reciprocating relationships with Hawkes processes (2012)
and 27 more
  1. Stability of nonlinear Hawkes processes (1996)
  2. Power spectra of random spike fields and related processes (2005)
  3. Mean-field limit of generalized Hawkes processes (2017)
  4. Inferring the dependence graph density of binary graphical models in high dimension (2024)
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0 of 35 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.