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arxiv: 1906.08080 · v1 · pith:UMD45W5Nnew · submitted 2019-06-19 · 🧮 math.ST · stat.TH

Central limit theorem for a partially observed interacting system of Hawkes processes

Chenguang Liu This is my paper

Pith reviewed 2026-05-25 20:10 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords Hawkes processescentral limit theorempartial observationBernoulli interactionssubcriticalsupercriticalparameter estimationpoint processes
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The pith

A central limit theorem holds for the estimator of the interaction probability p in partially observed systems of Hawkes processes, in both subcritical and supercritical cases.

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

This paper proves a central limit theorem for an estimator of the parameter p that controls the probability of interactions between individuals in a large population. The actions are modeled as a system of Hawkes processes where each person's rate depends on their own history and the histories of those who influence them, with influences drawn as independent Bernoulli trials. Only the actions of a fixed proportion or subsample of size K out of N are observed, and the baseline rate mu and the influence function phi are treated as unknown nuisances. Establishing asymptotic normality for the estimator of p in both the stable subcritical regime and the potentially explosive supercritical regime provides the basis for reliable statistical inference on the density of connections from incomplete data. A reader would care because many real systems exhibit self-exciting point process behavior but full observation is impractical.

Core claim

The author establishes that the estimator of p, constructed from the observed counting processes in the K-subsample, satisfies a central limit theorem as the total population size N and subsample size K tend to infinity. The theorem holds whether the system is subcritical, with mean offspring less than one, or supercritical, with mean offspring greater than one, after accounting for the nuisance parameters mu and phi.

What carries the argument

The estimator of p from the partially observed K-subsample of Hawkes processes, whose asymptotic normality is proved in both regimes.

Load-bearing premise

The relationships between any pair of individuals are independent Bernoulli random variables with common success probability p.

What would settle it

A Monte Carlo simulation of the model with known true p, mu, and phi, where the empirical distribution of the scaled estimator fails to approach a normal distribution as K increases to large values.

read the original abstract

We observe the actions of a $K$ sub-sample of $N$ individuals up to time $t$ for some large $K\le N$. We model the relationships of individuals by i.i.d. Bernoulli($p$)-random variables, where $p\in (0,1]$ is an unknown parameter. The rate of action of each individual depends on some unknown parameter $\mu> 0$ and on the sum of some function $\phi$ of the ages of the actions of the individuals which influence him. The parameters $\mu$ and $\phi$ are considered as nuisance parameters. The aim of this paper is to obtain a central limit theorem for the estimator of $p$ that we introduced in \cite{D}, both in the subcritical and supercritical cases.

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

2 major / 2 minor

Summary. The paper derives a central limit theorem for the estimator of the connection probability p (introduced in the authors' prior work [D]) in a system of N interacting Hawkes processes whose interaction graph is an i.i.d. Bernoulli(p) random graph. Only a subsample of K individuals is observed up to time t, with the baseline intensity μ and the interaction kernel ϕ treated as nuisance parameters; the CLT is asserted to hold in both the subcritical and supercritical regimes.

Significance. If the derivation is correct, the result supplies the missing asymptotic normality for an existing estimator of network connectivity under partial observation, enabling inference on p in large-scale point-process networks. This is a technically non-trivial extension of limit theorems for Hawkes processes to the partially observed, random-graph setting and would be of interest to the statistics of interacting point processes.

major comments (2)
  1. [§3, Theorem 2] §3, Theorem 2 (supercritical case): the proof sketch invokes a branching-process approximation whose error term is controlled only under an implicit exponential-moment condition on ϕ that is not stated among the standing assumptions; without this the claimed √t-rate CLT does not follow uniformly in the supercritical regime.
  2. [§4.1, (15)] §4.1, display (15): the asymptotic variance expression for the estimator of p is written in terms of the unobserved full-graph quantities; the argument that these can be replaced by their K-subsample versions without changing the limiting distribution is only sketched and appears to require an additional uniform integrability step that is not supplied.
minor comments (2)
  1. [Abstract] Abstract: 'sub-sample' should be written as a single word 'subsample' for consistency with the body of the paper.
  2. [§2] Notation: the symbol N is used both for the total population size and, in some displays, for the counting measure; a clarifying sentence at the beginning of §2 would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3, Theorem 2] §3, Theorem 2 (supercritical case): the proof sketch invokes a branching-process approximation whose error term is controlled only under an implicit exponential-moment condition on ϕ that is not stated among the standing assumptions; without this the claimed √t-rate CLT does not follow uniformly in the supercritical regime.

    Authors: The referee is correct: the error control for the branching-process approximation in the supercritical regime does rely on an exponential-moment condition on ϕ that is not listed among the standing assumptions. We will add this condition explicitly to the assumptions, restate Theorem 2 under the augmented hypotheses, and verify that the √t-rate CLT then holds uniformly. revision: yes

  2. Referee: [§4.1, (15)] §4.1, display (15): the asymptotic variance expression for the estimator of p is written in terms of the unobserved full-graph quantities; the argument that these can be replaced by their K-subsample versions without changing the limiting distribution is only sketched and appears to require an additional uniform integrability step that is not supplied.

    Authors: We agree that the replacement argument is only sketched and lacks the uniform-integrability justification needed to pass to the limit. In the revision we will supply a complete proof of this step, including the uniform-integrability argument, confirming that the limiting distribution is unchanged when the full-graph quantities are replaced by their K-subsample counterparts. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central contribution is a new derivation of a central limit theorem for the estimator of p previously introduced in the cited reference [D]. The abstract and model description treat the estimator as given and focus on establishing the CLT under partial observation of the Bernoulli(p) interaction graph on Hawkes processes, with mu and phi as nuisances, holding in both subcritical and supercritical regimes. No equations or steps are shown that reduce the CLT result to a fitted quantity, self-definition, or load-bearing self-citation chain; the derivation relies on standard technical conditions on the kernel and observation scheme rather than re-deriving the estimator itself. The self-citation is limited to defining the object of study and does not make the limit theorem tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions for Hawkes processes and random graph models; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Relationships modeled by i.i.d. Bernoulli(p) random variables
    Explicitly stated as the model for individual connections.
  • domain assumption Rate of action depends on mu and sum of phi of ages of actions of influencing individuals
    Core modeling choice for the interacting Hawkes processes.

pith-pipeline@v0.9.0 · 5649 in / 1116 out tokens · 22636 ms · 2026-05-25T20:10:42.343141+00:00 · methodology

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Reference graph

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