• Bernoulli
  • Volume 25, Number 3 (2019), 1724-1754.

Mixing properties and central limit theorem for associated point processes

Arnaud Poinas, Bernard Delyon, and Frédéric Lavancier

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Positively (resp. negatively) associated point processes are a class of point processes that induce attraction (resp. inhibition) between the points. As an important example, determinantal point processes (DPPs) are negatively associated. We prove $\alpha $-mixing properties for associated spatial point processes by controlling their $\alpha $-coefficients in terms of the first two intensity functions. A central limit theorem for functionals of associated point processes is deduced, using both the association and the $\alpha $-mixing properties. We discuss in detail the case of DPPs, for which we obtain the limiting distribution of sums, over subsets of close enough points of the process, of any bounded function of the DPP. As an application, we get the asymptotic properties of the parametric two-step estimator of some inhomogeneous DPPs.

Article information

Bernoulli, Volume 25, Number 3 (2019), 1724-1754.

Received: May 2017
Revised: February 2018
First available in Project Euclid: 12 June 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

determinantal point process negative association parametric estimation positive association strong mixing


Poinas, Arnaud; Delyon, Bernard; Lavancier, Frédéric. Mixing properties and central limit theorem for associated point processes. Bernoulli 25 (2019), no. 3, 1724--1754. doi:10.3150/18-BEJ1033.

Export citation


  • [1] Alam, K. and Saxena, K.M.L. (1981). Positive dependence in multivariate distributions. Comm. Statist. Theory Methods 10 1183–1196.
  • [2] Bardenet, R. and Hardy, A. (2016). Monte Carlo with determinantal point processes. ArXiv preprint. Available at arXiv:1605.00361.
  • [3] Biscio, C.A.N. and Lavancier, F. (2016). Brillinger mixing of determinantal point processes and statistical applications. Electron. J. Stat. 10 582–607.
  • [4] Biscio, C.A.N. and Lavancier, F. (2016). Quantifying repulsiveness of determinantal point processes. Bernoulli 22 2001–2028.
  • [5] Biscio, C.A.N., Poinas, A. and Waagepetersen, R. (2018). A note on gaps in proofs of central limit theorems. Statist. Probab. Lett. 135 7–10.
  • [6] Błaszczyszyn, B. and Yogeshwaran, D. (2015). Clustering comparison of point processes, with applications to random geometric models. In Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Math. 2120 31–71. Cham: Springer.
  • [7] Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. Ann. Probab. 10 1047–1050.
  • [8] Bradley, R.C. (2005). Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surv. 2 107–144. Update of, and a supplement to, the 1986 original.
  • [9] Bulinski, A. and Shashkin, A. (2007). Limit Theorems for Associated Random Fields and Related Systems. Advanced Series on Statistical Science & Applied Probability 10. Hackensack, NJ: World Scientific Co. Pte. Ltd.
  • [10] Bulinski, A.V. and Shabanovich, È. (1998). Asymptotic behavior of some functionals of positively and negatively dependent random fields. Fundam. Prikl. Mat. 4 479–492.
  • [11] Burton, R. and Waymire, E. (1985). Scaling limits for associated random measures. Ann. Probab. 13 1267–1278.
  • [12] Daley, D.J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Vol. I, 2nd ed. Probability and Its Applications (New York). New York: Springer. Elementary theory and methods.
  • [13] Davydov, J.A. (1968). The convergence of distributions which are generated by stationary random processes. Teor. Veroyatn. Primen. 13 730–737.
  • [14] Dellacherie, C. and Meyer, P.-A. (1978). Probabilities and Potential. North-Holland Mathematics Studies 29. Amsterdam: North-Holland.
  • [15] Deng, N., Zhou, W. and Haenggi, M. (2015). The Ginibre point process as a model for wireless networks with repulsion. IEEE Trans. Wirel. Commun. 1 479–492.
  • [16] Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statistics 85. New York: Springer.
  • [17] Doukhan, P., Fokianos, K. and Li, X. (2012). On weak dependence conditions: The case of discrete valued processes. Statist. Probab. Lett. 82 1941–1948.
  • [18] Doukhan, P. and Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities. Stochastic Process. Appl. 84 313–342.
  • [19] Esary, J.D., Proschan, F. and Walkup, D.W. (1967). Association of random variables, with applications. Ann. Math. Stat. 38 1466–1474.
  • [20] Evans, S.N. (1990). Association and random measures. Probab. Theory Related Fields 86 1–19.
  • [21] Ghosh, S. (2015). Determinantal processes and completeness of random exponentials: The critical case. Probab. Theory Related Fields 163 643–665.
  • [22] Gomez, J.S., Vasseur, A., Vergne, A., Martins, P., Decreusefond, L. and Chen, W. (2015). A case study on regularity in cellular network deployment. IEEE Wirel. Commun. Lett. 4 421–424.
  • [23] Guyon, X. (1995). Random Fields on a Network. Probability and Its Applications (New York). New York: Springer. Modeling, statistics, and applications, Translated from the 1992 French original by Carenne Ludeña.
  • [24] Heinrich, L. (2016). On the strong Brillinger-mixing property of $\alpha$-determinantal point processes and some applications. Appl. Math. 61 443–461.
  • [25] Heinrich, L. and Klein, S. (2014). Central limit theorems for empirical product densities of stationary point processes. Stat. Inference Stoch. Process. 17 121–138.
  • [26] Hough, J.B., Krishnapur, M., Peres, Y. and Virág, B. (2009). Zeros of Gaussian Analytic Functions and Determinantal Point Processes. University Lecture Series 51. Providence, RI: Amer. Math. Soc.
  • [27] Ibragimov, I.A. and Linnik, Y.V. (1971). Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Noordhoff Publishing. With a supplementary chapter by I.A. Ibragimov and V.V. Petrov, Translation from the Russian edited by J. F. C. Kingman.
  • [28] Jolivet, E. (1981). Central limit theorem and convergence of empirical processes for stationary point processes. In Point Processes and Queuing Problems (Colloq., Keszthely, 1978). Colloquia Mathematica Societatis János Bolyai 24 117–161. Amsterdam: North-Holland.
  • [29] Kulesza, A. and Taskar, B. (2012). Determinantal point process models for machine learning. Found. Trends Mach. Learn. 5 123–286.
  • [30] Lavancier, F., Møller, J. and Rubak, E. (2015). Determinantal point process models and statistical inference. J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 853–877.
  • [31] Lyons, R. (2014). Determinantal probability: Basic properties and conjectures. In Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. IV 137–161. Seoul: Kyung Moon Sa.
  • [32] Macchi, O. (1975). The coincidence approach to stochastic point processes. Adv. in Appl. Probab. 7 83–122.
  • [33] Miyoshi, N. and Shirai, T. (2014). A cellular network model with Ginibre configured base stations. Adv. in Appl. Probab. 46 832–845.
  • [34] Møller, J. and Waagepetersen, R.P. (2004). Statistical Inference and Simulation for Spatial Point Processes. Monographs on Statistics and Applied Probability 100. Boca Raton, FL: Chapman & Hall/CRC.
  • [35] Rio, E. (1993). Covariance inequalities for strongly mixing processes. Ann. Inst. Henri Poincaré Probab. Stat. 29 587–597.
  • [36] Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 43–47.
  • [37] 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.
  • [38] Soshnikov, A. (2000). Determinantal random point fields. Uspekhi Mat. Nauk 55 107–160.
  • [39] Waagepetersen, R. and Guan, Y. (2009). Two-step estimation for inhomogeneous spatial point processes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 71 685–702.
  • [40] Yuan, M., Su, C. and Hu, T. (2003). A central limit theorem for random fields of negatively associated processes. J. Theoret. Probab. 16 309–323.