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arxiv: 2606.27898 · v1 · pith:4KHT5EWRnew · submitted 2026-06-26 · 🧮 math.PR

Quantitative CLTs for Geometric Statistics of Dependent Marked Point Processes

Tianshu Cong , Aihua Xia , J. E. Yukich This is my paper

Pith reviewed 2026-06-29 03:22 UTC · model grok-4.3

classification 🧮 math.PR
keywords central limit theoremWasserstein distancemarked point processesgeometric statisticsmixing conditionsnormal approximationdependent marksweak dependence
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The pith

Weak mixing conditions based on geometric criteria deliver Wasserstein rates of normal approximation for geometric statistics on marked point processes with dependent marks.

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

The paper extends prior central limit theorems for geometric statistics of point processes to the setting of marked processes whose marks exhibit dependence. It supplies verifiable geometric mixing conditions that, together with fast decay of correlations and variance growth, control the dependence enough to produce explicit rates of convergence to normality in the Wasserstein metric. These conditions apply to a broad class that includes determinantal point processes, unevenly spaced time series, continuum percolation, and local U-statistics on spatial graphs. A sympathetic reader cares because the results turn abstract dependence assumptions into checkable geometric tests while furnishing quantitative error bounds rather than mere existence of a limit.

Core claim

We establish weak mixing conditions yielding rates of normal convergence in the Wasserstein distance for geometric statistics of a large class of point processes with dependent marks. The mixing conditions are expressed in terms of verifiable geometric criteria and are shown to be sufficient when the underlying marked point process satisfies fast decay of correlations together with a variance growth condition.

What carries the argument

Verifiable geometric mixing criteria that bound the dependence among local scores of the statistic, thereby controlling the Wasserstein distance to the normal law.

If this is right

  • Quantitative normal approximation holds for local U-statistics of determinantal point processes with dependent marks.
  • Rates apply to statistics of continuum percolation clusters and to interacting diffusions on spatial random graphs.
  • The same geometric criteria yield Wasserstein bounds for unevenly spaced time series.
  • The mixing conditions are strictly weaker than the dependence assumptions used in earlier work on unmarked processes.

Where Pith is reading between the lines

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

  • The geometric criteria could be checked numerically on simulated realizations to certify applicability before using the normal approximation in practice.
  • The same mixing framework may extend to other distances such as Kolmogorov or total variation once the Stein-method machinery is adjusted.
  • The results suggest that dependence in spatial marked data can be controlled by local geometric separation rather than global correlation decay alone.

Load-bearing premise

The marked point process must satisfy fast decay of correlations and variance growth, and the new geometric mixing criteria must be strong enough to dominate the dependence among the local scores.

What would settle it

A concrete marked point process satisfying the stated geometric mixing conditions for which the Wasserstein distance between the normalized statistic and the normal law fails to decay at the rate predicted by the bound.

Figures

Figures reproduced from arXiv: 2606.27898 by Aihua Xia, J. E. Yukich, Tianshu Cong.

Figure 1
Figure 1. Figure 1: Interval partition of Γλ for polynomial mixing The renewal points in the stationary renewal point process P on a given interval I are simply the points of the process P that fall within the interval I, expressed as P|I . From Lemma 3.7 (with t and C replaced by ρ/3 and 1, respectively) and extending the probability space if necessary, we can construct, on the same probability space, a copy {P′ |B1,e ,P ′ |… view at source ↗
read the original abstract

Given a geometric statistic expressible as a sum of scores which depend on local data, \citet{BYY19} established central limit theorems for centered and normalized versions of these statistics, subject to the underlying point process having fast decay of correlations and also subject to a variance growth condition. Building on this, \citet{CX23} derived rates of normal approximation, as measured by the Wasserstein distance, for statistics of point processes exhibiting fast decay of dependence. Here we go further and establish weak mixing conditions yielding rates of normal convergence for geometric statistics of marked point processes. The mixing conditions, which are in terms of verifiable geometric criteria, provide rates of normal approximation in the Wasserstein distance for statistics of a large class of point processes with dependent marks. Examples include statistics of determinantal point processes, unevenly spaced time series, continuum percolation, interacting diffusions on spatial random graphs, as well as local $U$-statistics.

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 manuscript claims that weak mixing conditions, formulated via verifiable geometric criteria, yield rates of normal convergence in the Wasserstein distance for geometric statistics (expressible as sums of local scores) of marked point processes with dependent marks. This extends the CLTs of BYY19 (under fast decay of correlations and variance growth) and the Wasserstein bounds of CX23, and is illustrated with examples including determinantal point processes, unevenly spaced time series, continuum percolation, interacting diffusions on spatial random graphs, and local U-statistics.

Significance. If the derivations hold, this provides a useful extension of quantitative CLTs to marked point processes under geometrically checkable mixing conditions, broadening applicability in spatial statistics, stochastic geometry, and dependent data settings. Credit is due for the emphasis on verifiable criteria and the concrete list of examples, which help demonstrate utility without introducing circularity in the inherited variance-growth assumption from the cited works.

minor comments (2)
  1. [Abstract] The abstract could state the explicit form of the Wasserstein rate (e.g., dependence on mixing coefficients or dimension) rather than only describing its existence.
  2. [§2] In the setup of the marked point process and geometric statistic (likely §2), ensure the notation for marks and local scores is defined before first use in the main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of extending quantitative CLTs to marked point processes under verifiable geometric mixing conditions, and the recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; extends prior independent results via new verifiable criteria

full rationale

The derivation explicitly builds the new weak-mixing criteria and Wasserstein rates on the base CLT and variance-growth conditions from the cited BYY19 and CX23. These citations supply the starting CLT and variance assumption, while the paper's contribution (geometric mixing conditions in terms of verifiable criteria) is presented as an independent extension applicable to marked processes. No self-definitional reduction, fitted-input prediction, or load-bearing self-citation chain is exhibited in the abstract or described structure; the mixing-to-bound step is claimed to be controlled by the new criteria under the inherited assumptions. This is the normal case of incremental extension rather than circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters or invented entities; the work relies on standard measure-theoretic probability and the mixing framework of the cited papers.

axioms (2)
  • standard math Standard axioms of probability measures on marked point processes
    Invoked implicitly to define the underlying probability space and expectations of the geometric statistics.
  • domain assumption Fast decay of correlations and variance growth condition from BYY19
    Required for the base central limit theorem before the new mixing rates are applied.

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Reference graph

Works this paper leans on

46 extracted references · 5 canonical work pages · 2 internal anchors

  1. [1]

    and Hammersley, J

    Beardwood, J., Halton, J. and Hammersley, J. (1959). The shortest path through many points.Math. Proc. Camb. Philos. Soc.55, 299–327

    1959

  2. [2]

    and Večeřa, J

    Beneš, V., Hofer-Temmel, C., Last, G. and Večeřa, J. (2020). Decorrelation of a class of Gibbs particle processes and asymptotic properties ofU-statistics.J. Appl. Prob.57, 928–955

    2020

  3. [3]

    and Breiman, L

    Bickel, P. and Breiman, L. (1983). Sums of functions of nearest neighbor distances, moment bounds, limit theorems, and a goodness of fit test.Ann. Prob.11, 184-214

    1983

  4. [4]

    and Yukich, J

    Błaszczyszyn, B., Yogeshwaran, D. and Yukich, J. E. (2019). Limit theory for geo- metric statistics of point processes having fast decay of correlations.Ann. Prob.47, 835–895

    2019

  5. [5]

    Limit theory for Lipschitz-localized statistics in random geometric models

    Błaszczyszyn, B., Yogeshwaran, D. and Yukich, J. E. (2026). Limit theory for Lipschitz-localized statistics in random geometric models. arXiv:2605.28430

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  6. [6]

    Bradley, R. C. (2005). Basic properties of strong mixing conditions. A survey and some open questions.Prob. Surveys2, 107–144

    2005

  7. [7]

    Brockwell, P. J. and Davis, R. A. (2016).Introduction to Time Series and Forecasting (3rd ed.). Springer

    2016

  8. [8]

    and Shashkin, A

    Bulinski, A. and Shashkin, A. (2007).Limit Theorems for Associated Random Fields and Related Systems.Vol. 10. World Scientific

    2007

  9. [9]

    Chen, L. H. Y., Goldstein, L. and Shao, Q. M. (2011).Normal Approximation By Stein’s Method. Springer-Verlag. 64

    2011

  10. [10]

    and Xia, A

    Cong, T. and Xia, A. (2023). Convergence rate for geometric statistics of point processes having fast decay of dependence.Electron. J. Prob.28, 1–35

    2023

  11. [11]

    and Xia, A

    Cong, T. and Xia, A. (2024). Normal approximation in total variation for statistics in geometric probability.Adv. Appl. Prob.56, 106–155

    2024

  12. [12]

    Davydov, Y. A. (1968). Convergence of Distributions Generated by Stationary Stochastic Processes.Theory Probab. Its Appl.13, 691–696

    1968

  13. [13]

    Notes in Math.,2237, 181–229

    Dereudre, D.(2019).IntroductiontothetheoryofGibbspointprocesses.InStochas- tic Geometry, Lect. Notes in Math.,2237, 181–229. Springer

    2019

  14. [14]

    and Tran, M.H

    Dinh, T.C., Ghosh, S., Tran, H.S. and Tran, M.H. (2025). Gaussian fluctuations for spinsystemsandpointprocesses: near-optimalratesviaquantitativeMarcinkiewicz’s theorem. arXiv:2107.08469v3

    work page arXiv 2025

  15. [15]

    and Thäle, C

    Eichelsbacher, P. and Thäle, C. (2014). New Berry-Esseen bounds for non-linear functionals of Poisson random measures.Electron. J. Probab.,19, 1–25

    2014

  16. [16]

    Forrester, P. J. and Lebowitz, J. L. (2014). Local central limit theorem for determi- nantal point processes.J. Stat. Phys.157, 60–69

    2014

  17. [17]

    Hamilton, J. D. (1994).Time Series Analysis. Princeton University Press

    1994

  18. [18]

    Hernández Ruíz, L. I. (2025). Results for convergence rates associated with renewal processes. InSéminaire de Probabilités LII,Lect. Notes in Math.,2363, 427–450. Springer

    2025

  19. [19]

    and Owada, T

    Hirsch, C., Lachièze-Rey, R. and Owada, T. (2025). Normal approximation for subgraph counts in age-dependent random connection models, arXiv:2505.09318

    work page arXiv 2025

  20. [20]

    and Svane, A

    Hirsch, C., Otto, M. and Svane, A. M. (2025). Normal approximation for Gibbs processes via disagreement couplings.Electron. J. Prob.30, 1–56

    2025

  21. [21]

    and Svane, A

    Hirsch, C., Otto, M. and Svane, A. M. (2026). Normal and Poisson approximation for Gibbs point processes with pair potentials. arXiv:2601.18695

    work page arXiv 2026

  22. [22]

    B., Krishnapur, M., Peres, Y

    Hough, J. B., Krishnapur, M., Peres, Y. and Viràg, B. (2009).Zeros of Gaussian an- alytic functions and determinantal point processes. American Mathematical Society, Providence, RI

    2009

  23. [23]

    (1983).Random Measures

    Kallenberg, O. (1983).Random Measures. Academic Press, London

    1983

  24. [24]

    and Yukich, J

    Lachièze-Rey, R., Schulte, M. and Yukich, J. E. (2019). Normal approximation for stabilizing functionals.Ann. Appl. Prob.29, 931–993

    2019

  25. [25]

    and Wu, R

    Lacker, D., Ramanan, K. and Wu, R. (2023). Local weak convergence for sparse networks of interacting processes.Ann. Appl. Prob.33, 843–888. 65

    2023

  26. [26]

    and Rubak, E

    Lavancier, F., Møller, J. and Rubak, E. (2015). Determinantal point process models and statistical inference.J. R. Stat. Soc., B: Stat. Methodol.77, 853–877

    2015

  27. [27]

    Lo, T. Y. Y. and Xia, A. (2024). On the rate of normal approximation for Poisson continuum percolation.Stat. Probab. Lett.210, Article 110110

    2024

  28. [28]

    and Roy, R

    Meester, R. and Roy, R. (1996).Continuum Percolation.Oxford University Press

    1996

  29. [29]

    and Perkins, W

    Michelen, M. and Perkins, W. (2022). Strong spatial mixing for repulsive point processes.J. Stat. Phys.189, Art. No. 9

    2022

  30. [30]

    and O’Reilly, E

    Møller, J. and O’Reilly, E. (2021). Couplings for determinantal point processes and their reduced Palm distributions with a view to quantifying repulsiveness.J. Appl. Prob.58, 469–483

    2021

  31. [31]

    Penrose, M. D. and Pisztora, A. (1996). Large deviations for discrete and continuous percolation.Adv. Appl. Prob.28, 29–52

    1996

  32. [32]

    and Schulte, M

    Reitzner, M. and Schulte, M. (2013). Central limit theorems forU-statistics of Poisson point processes.Ann. Prob.41, 3879–3909

    2013

  33. [33]

    Revuz, D. (1970). Mesures associées aux fonctionnelles additives de Markov. I. Trans. Amer. Math. Soc.148, 501–531

    1970

  34. [34]

    Rio, E. (1993). Covariance inequalities for strongly mixing processes.Ann. Inst. H. Poincaré Prob. Statist.29, 587–597

    1993

  35. [35]

    Rosenthal, H. P. (1970). On the subspaces ofLp(p >2)spanned by sequences of independent random variables.Israel J. Math.8, 273–303

    1970

  36. [36]

    (2010).Introduction to Probability Models (Tenth Edition)

    Ross, S. (2010).Introduction to Probability Models (Tenth Edition). Academic Press

    2010

  37. [37]

    and Yukich, J

    Schreiber, T. and Yukich, J. E. (2013). Limit theorems for geometric functionals of Gibbs point processes.Ann. Inst. H. Poincaré Prob. Statist.49, 1158–1182

    2013

  38. [38]

    and Yukich, J

    Schulte, M. and Yukich, J. E. (2023). Rates of multivariate normal approximation for statistics in geometric probability.Ann. Appl. Prob.,33, 507–548

    2023

  39. [39]

    and Takahashi, Y

    Shirai, T. and Takahashi, Y. (2003). Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes.J. Funct. Anal.205, 414–463

    2003

  40. [40]

    Soshnikov, A. (2002). Gaussian limit for determinantal random point fields.Ann. Prob.30, 171–187

    2002

  41. [41]

    Steele, J. M. (1981). Subadditive Euclidean functionals and nonlinear growth in geometric probability.Ann. Prob.9, 365–376. 66

    1981

  42. [42]

    (1990).Non-linear Time Series: A Dynamical System Approach

    Tong, H. (1990).Non-linear Time Series: A Dynamical System Approach. Oxford University Press

    1990

  43. [43]

    Second-order Poincar\'e inequalities and localization on the Poisson space

    Trauthwein, T. and Yukich, J. E. (2026). Second-order Poincaré inequalities and localization on the Poisson space. arXiv:2605.23292

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  44. [44]

    Wiroonsri, N. (2019). Normal approximation for associated point processes via Stein’s method with applications to determinantal point processes.J. Math. Anal. Appl.480, 123396

    2019

  45. [45]

    and Yukich, J

    Xia, A. and Yukich, J. E. (2015). Normal approximation for statistics of Gibbsian input in geometric probability.Adv. Appl. Prob.47, 934–972

    2015

  46. [46]

    Yukich, J. E. (2006). Ultra-small scale-free geometric networks,J. Appl. Prob.,43, 665–677. 67

    2006