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arxiv: 2605.28430 · v1 · pith:MGU4YSOWnew · submitted 2026-05-27 · 🧮 math.PR

Limit theory for Lipschitz-localized statistics in random geometric models

B. B{\l}aszczyszyn , D. Yogeshwaran , J. E. Yukich This is my paper

Pith reviewed 2026-06-29 10:22 UTC · model grok-4.3

classification 🧮 math.PR
keywords point processescentral limit theoremLipschitz localizationgeometric mixingmarked point processesstochastic geometryrandom geometric graphsspatial statistics
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The pith

Geometric mixing on point processes plus a Lipschitz localization condition on scores yields central limit theorems and asymptotic mean-variance formulas for their sums.

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

The paper develops a general limit theory for sums of locally dependent scores attached to marked Euclidean point processes. It replaces classical stabilization via stopping sets with a distributional Lipschitz-localization condition defined through the bounded Lipschitz metric on the scores. Geometric mixing assumptions on the point process then guarantee a central limit theorem together with explicit asymptotics for the expectation and variance. The same conditions apply to several concrete models including spin systems on spatial random graphs, interacting diffusions, interacting particle systems, empirical random fields, and geostatistical Boolean models.

Core claim

Geometric mixing conditions on the underlying point process and a Lipschitz-localization condition on the scores, formulated using the bounded Lipschitz metric as a distributional criterion, jointly ensure a central limit theorem for the sums of the scores as well as expectation and variance asymptotics.

What carries the argument

The Lipschitz-localization condition on the scores, defined via the bounded Lipschitz metric, which supplies a distributional criterion that controls local dependence without stopping-set constructions.

If this is right

  • Expectation and variance of the sums admit explicit asymptotic expressions under the stated conditions.
  • The central limit theorem holds for the total score in spin systems, interacting diffusions, and interacting particle systems on random geometric graphs.
  • The same limit theory covers empirical random fields and geostatistical Boolean models.
  • The framework supplies an alternative route to limit theorems that avoids explicit construction of stabilizing stopping sets.

Where Pith is reading between the lines

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

  • The bounded-Lipschitz formulation may allow direct verification of the condition in models where stopping-set radii are hard to control explicitly.
  • Similar mixing-plus-localization arguments could be tested on marked point processes defined on non-Euclidean spaces or on graphs with different geometry.
  • The approach suggests checking the Lipschitz condition first when adapting existing stabilization proofs to new interacting particle systems.

Load-bearing premise

The scores satisfy the Lipschitz-localization condition formulated with the bounded Lipschitz metric.

What would settle it

A concrete marked point process and score function that obey the geometric mixing and Lipschitz-localization conditions yet whose properly normalized sum fails to converge in distribution to a Gaussian would falsify the central limit theorem.

read the original abstract

We study sums of locally dependent scores associated with general marked (i.e., labeled) Euclidean point processes. We introduce geometric mixing conditions on the underlying point process and a Lipschitz-"localization" condition on the scores, which jointly ensure a central limit theorem for the sums of the scores as well as expectation and variance asymptotics. Our localization condition is formulated using the bounded Lipschitz metric, providing a distributional criterion. This stands in contrast to the classical stabilization conditions in stochastic geometry, which are typically based on stopping-set constructions. To demonstrate the applicability of our general framework, we consider several stochastic processes indexed by spatial random graphs. These include spin systems, interacting diffusions, and interacting particle systems. In particular, spin systems highlight the importance of our localization condition. Additional applications include empirical random fields and geostatistical Boolean models.

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 develops limit theorems for sums of scores on marked Euclidean point processes. It posits geometric mixing conditions on the underlying point process together with a Lipschitz-localization condition on the scores (formulated via the bounded Lipschitz metric on probability measures) and claims that these two conditions jointly imply a central limit theorem for the sums as well as asymptotic expressions for the mean and variance. The framework is illustrated on spin systems, interacting diffusions, interacting particle systems, empirical random fields, and geostatistical Boolean models.

Significance. If the central claims are valid, the work supplies a distributional alternative to classical stabilization arguments in stochastic geometry. The bounded-Lipschitz formulation may be easier to verify in certain dependent systems (e.g., spin systems) where stopping-set constructions are cumbersome. The explicit applications to several classes of spatial random graphs constitute a concrete strength.

major comments (2)
  1. [Abstract and localization-condition paragraph] Abstract and the paragraph introducing the localization condition: the bounded Lipschitz metric metrizes weak convergence, yet the manuscript does not demonstrate that this condition alone (or jointly with geometric mixing) yields uniform integrability or finite second moments of the localized scores. Variance asymptotics and any Lindeberg-type condition for the CLT require E[score²] < ∞; if these moment bounds are tacitly assumed rather than derived, the joint-sufficiency claim is incomplete.
  2. [Main theorem section] Section on the main theorem (presumably Theorem 3.1 or equivalent): the statement that geometric mixing plus Lipschitz-localization suffice for both the CLT and the variance asymptotics needs an explicit verification that the localization condition controls the second-moment tails, or else an additional moment hypothesis must be added to the theorem statement.
minor comments (2)
  1. [Abstract] Notation for the bounded Lipschitz metric should be introduced once and used consistently; the current abstract paragraph switches between descriptive phrases and symbols without a single defining equation.
  2. [Applications section] The applications section would benefit from a short table summarizing which processes satisfy the localization condition by direct verification versus those that require additional arguments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to clarify moment conditions. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and localization-condition paragraph] Abstract and the paragraph introducing the localization condition: the bounded Lipschitz metric metrizes weak convergence, yet the manuscript does not demonstrate that this condition alone (or jointly with geometric mixing) yields uniform integrability or finite second moments of the localized scores. Variance asymptotics and any Lindeberg-type condition for the CLT require E[score²] < ∞; if these moment bounds are tacitly assumed rather than derived, the joint-sufficiency claim is incomplete.

    Authors: We agree that the bounded Lipschitz metric metrizes weak convergence and does not by itself guarantee finite second moments or uniform integrability. The localization condition is distributional, and while it is used jointly with geometric mixing to obtain the CLT, we acknowledge that finite second moments of the scores must be assumed separately for the variance asymptotics and Lindeberg condition to hold. In the revision we will add an explicit second-moment hypothesis to the main theorem statement, update the abstract and introduction to reflect this, and note that the hypothesis is verified directly in each application. revision: yes

  2. Referee: [Main theorem section] Section on the main theorem (presumably Theorem 3.1 or equivalent): the statement that geometric mixing plus Lipschitz-localization suffice for both the CLT and the variance asymptotics needs an explicit verification that the localization condition controls the second-moment tails, or else an additional moment hypothesis must be added to the theorem statement.

    Authors: We concur that the localization condition does not control second-moment tails. We will therefore revise the theorem to include the additional hypothesis that the scores possess uniformly bounded second moments (rather than attempting to derive such bounds from localization alone). A short remark will be added explaining the necessity of this hypothesis and confirming that it holds in the listed applications. revision: yes

Circularity Check

0 steps flagged

No circularity; external mixing and localization conditions independently imply the limit theorems.

full rationale

The paper states that geometric mixing conditions on the point process together with a Lipschitz-localization condition (formulated via the bounded Lipschitz metric) jointly ensure the CLT plus expectation and variance asymptotics. These are introduced as assumptions contrasting with stabilization, with no equations, fitted parameters, or self-citations shown to reduce the claimed results to the inputs by construction. No self-definitional steps, fitted-input predictions, or load-bearing self-citation chains appear in the provided text. The framework is therefore self-contained against external benchmarks, consistent with a normal non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on geometric mixing (domain assumption) and the new Lipschitz-localization condition (ad_hoc_to_paper). No free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption Geometric mixing conditions on the point process ensure dependence decays with distance
    Invoked to guarantee the CLT holds for the sums.
  • ad hoc to paper Scores satisfy Lipschitz-localization in the bounded Lipschitz metric
    Central new assumption introduced to replace stabilization.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Quantitative CLTs for Geometric Statistics of Dependent Marked Point Processes

    math.PR 2026-06 unverdicted novelty 5.0

    Derives Wasserstein bounds on normal approximation for sums of local scores in marked point processes with dependent marks via geometric weak mixing conditions.

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