The Annals of Probability

Limit theory for geometric statistics of point processes having fast decay of correlations

B. Błaszczyszyn, D. Yogeshwaran, and J. E. Yukich

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Abstract

Let $\mathcal{P}$ be a simple, stationary point process on $\mathbb{R}^{d}$ having fast decay of correlations, that is, its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let $\mathcal{P}_{n}:=\mathcal{P}\cap W_{n}$ be its restriction to windows $W_{n}:=[-{\frac{1}{2}}n^{1/d},{\frac{1}{2}}n^{1/d}]^{d}\subset\mathbb{R}^{d}$. We consider the statistic $H_{n}^{\xi}:=\sum_{x\in\mathcal{P}_{n}}\xi(x,\mathcal{P}_{n})$ where $\xi(x,\mathcal{P}_{n})$ denotes a score function representing the interaction of $x$ with respect to $\mathcal{P}_{n}$. When $\xi$ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics and central limit theorems for $H_{n}^{\xi}$ and, more generally, for statistics of the re-scaled, possibly signed, $\xi$-weighted point measures $\mu_{n}^{\xi}:=\sum_{x\in\mathcal{P}_{n}}\xi(x,\mathcal{P}_{n})\delta_{n^{-1/d}x}$, as $W_{n}\uparrow\mathbb{R}^{d}$. This gives the limit theory for nonlinear geometric statistics (such as clique counts, the number of Morse critical points, intrinsic volumes of the Boolean model and total edge length of the $k$-nearest neighbors graph) of $\alpha$-determinantal point processes (for $-1/\alpha\in\mathbb{N}$) having fast decreasing kernels, including the $\beta$-Ginibre ensembles, extending the Gaussian fluctuation results of Soshnikov [Ann. Probab. 30 (2002) 171–187] to nonlinear statistics. It also gives the limit theory for geometric $U$-statistics of $\alpha$-permanental point processes (for $1/\alpha\in\mathbb{N}$) as well as the zero set of Gaussian entire functions, extending the central limit theorems of Nazarov and Sodin [Comm. Math. Phys. 310 (2012) 75–98] and Shirai and Takahashi [J. Funct. Anal. 205 (2003) 414–463], which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in [Stochastic Process. Appl. 56 (1995) 321–335; Statist. Probab. Lett. 36 (1997) 299–306] to show the fast decay of the correlations of $\xi$-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing, and consequently yields the asymptotic normality of $\mu_{n}^{\xi}$ via an extension of the cumulant method.

Article information

Source
Ann. Probab., Volume 47, Number 2 (2019), 835-895.

Dates
Received: July 2016
Revised: January 2018
First available in Project Euclid: 26 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1551171639

Digital Object Identifier
doi:10.1214/18-AOP1273

Mathematical Reviews number (MathSciNet)
MR3916936

Zentralblatt MATH identifier
07053558

Subjects
Primary: 60F05: Central limit and other weak theorems 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G55: Point processes 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 05C80: Random graphs [See also 60B20]

Keywords
Point processes having fast decay of correlations determinantal point process permanental point process Gaussian entire functions Gibbs’ point process $U$-statistics stabilization difference operators cumulants Brillinger mixing central limit theorem

Citation

Błaszczyszyn, B.; Yogeshwaran, D.; Yukich, J. E. Limit theory for geometric statistics of point processes having fast decay of correlations. Ann. Probab. 47 (2019), no. 2, 835--895. doi:10.1214/18-AOP1273. https://projecteuclid.org/euclid.aop/1551171639


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References

  • [1] Baddeley, A. (1980). A limit theorem for statistics of spatial data. Adv. in Appl. Probab. 12 447–461.
  • [2] Barbour, A. D. and Xia, A. (2006). Normal approximation for random sums. Adv. in Appl. Probab. 38 693–728.
  • [3] Baryshnikov, Y., Eichelsbacher, P., Schreiber, T. and Yukich, J. E. (2008). Moderate deviations for some point measures in geometric probability. Ann. Inst. Henri Poincaré Probab. Stat. 44 422–446.
  • [4] Baryshnikov, Yu. and Yukich, J. E. (2003). Gaussian fields and random packing. J. Stat. Phys. 111 443–463.
  • [5] Baryshnikov, Yu. and Yukich, J. E. (2005). Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 213–253.
  • [6] Baumann, K. and Hegerfeldt, G. C. (1985). A noncommutative Marcinkiewicz theorem. Publ. Res. Inst. Math. Sci. 21 191–204.
  • [7] Biscio, C. A. N. and Lavancier, F. (2016). Brillinger mixing of determinantal point processes and statistical applications. Electron. J. Stat. 10 582–607.
  • [8] Błaszczyszyn, B. (1995). Factorial moment expansion for stochastic systems. Stochastic Process. Appl. 56 321–335.
  • [9] Błaszczyszyn, B., Merzbach, E. and Schmidt, V. (1997). A note on expansion for functionals of spatial marked point processes. Statist. Probab. Lett. 36 299–306.
  • [10] Błaszczyszyn, B., Yogeshwaran, D. and Yukich, J. E. (2018). Limit theory for geometric statistics of point processes having fast decay of correlations. Available at arXiv:1606.03988v3.
  • [11] Błaszczyszyn, B., Yogeshwaran, D. and Yukich, J. E. (2019). Supplement to “Limit theory for geometric statistics of point processes having fast decay of correlations.” DOI:10.1214/18-AOP1273SUPP.
  • [12] Bulinski, A. and Spodarev, E. (2013). Central limit theorems for weakly dependent random fields. In Stochastic Geometry, Spatial Statistics and Random Fields (E. Spodarev, ed.). Lecture Notes in Math. 2068 337–383. Springer, Heidelberg.
  • [13] Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I: Elementary Theory and Methods. Probability and Its Applications. Springer, New York.
  • [14] Eichelsbacher, P., Raič, M. and Schreiber, T. (2015). Moderate deviations for stabilizing functionals in geometric probability. Ann. Inst. Henri Poincaré Probab. Stat. 51 89–128.
  • [15] Eichelsbacher, P. and Thäle, C. (2014). New Berry–Esseen bounds for non-linear functionals of Poisson random measures. Electron. J. Probab. 19 no. 102, 25.
  • [16] Forrester, P. J. and Honner, G. (1999). Exact statistical properties of the zeros of complex random polynomials. J. Phys. A 32 2961–2981.
  • [17] Goldman, A. (2010). The Palm measure and the Voronoi tessellation for the Ginibre process. Ann. Appl. Probab. 20 90–128.
  • [18] Grote, J. and Thäle, C. (2016). Gaussian polytopes: A cumulant-based approach. Preprint. Available at arXiv:1602.06148.
  • [19] Hall, P. (1988). Introduction to the Theory of Coverage Processes. Wiley, New York.
  • [20] Hanisch, K.-H. (1982). On inversion formulae for $n$-fold Palm distributions of point processes in LCS-spaces. Math. Nachr. 106 171–179.
  • [21] Heinrich, L. (1994). Normal approximation for some mean-value estimates of absolutely regular tessellations. Math. Methods Statist. 3 1–24.
  • [22] Heinrich, L. (2013). Asymptotic methods in statistics of random point processes. In Stochastic Geometry, Spatial Statistics and Random Fields (E. Spodarev, ed.). Lecture Notes in Math. 2068 115–150. Springer, Heidelberg.
  • [23] Heinrich, L. and Molchanov, I. S. (1999). Central limit theorem for a class of random measures associated with germ-grain models. Adv. in Appl. Probab. 31 283–314.
  • [24] Hough, J. B., Krishnapur, M., Peres, Y. and Virág, B. (2009). Zeros of Gaussian Analytic Functions and Determinantal Point Processes. University Lecture Series 51. Amer. Math. Soc., Providence, RI.
  • [25] Ivanoff, G. (1982). Central limit theorems for point processes. Stochastic Process. Appl. 12 171–186.
  • [26] Krishnapur, M. (2006). Overcrowding estimates for zeroes of planar and hyperbolic Gaussian analytic functions. J. Stat. Phys. 124 1399–1423.
  • [27] Lachièze-Rey, R. and Peccati, G. (2017). New Berry–Esseen bounds for functionals of binomial point processes. Ann. Appl. Probab. 27 1992–2031.
  • [28] Lachièze-Rey, R., Schulte, M. and Yukich, J. E. (2017). Normal approximation for stabilizing functionals. Preprint. Available at arXiv:1702.00726.
  • [29] Last, G., Peccati, G. and Schulte, M. (2016). Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Probab. Theory Related Fields 165 667–723.
  • [30] Last, G. and Penrose, M. D. (2011). Poisson process Fock space representation, chaos expansion and covariance inequalities. Probab. Theory Related Fields 150 663–690.
  • [31] Last, G. and Penrose, M. D. (2013). Percolation and limit theory for the Poisson lilypond model. Random Structures Algorithms 42 226–249.
  • [32] Malyšev, V. A. (1975). A central limit theorem for Gibbsian random fields. Soviet Math. Doklady 16 1141–1145.
  • [33] Malyshev, V. A. and Minlos, R. A. (1991). Gibbs Random Fields: Cluster Expansions. Mathematics and Its Applications (Soviet Series) 44. Kluwer Academic, Dordrecht.
  • [34] Martin, P. A. and Yalcin, T. (1980). The charge fluctuations in classical Coulomb systems. J. Stat. Phys. 22 435–463.
  • [35] McCullagh, P. and Møller, J. (2006). The permanental process. Adv. in Appl. Probab. 38 873–888.
  • [36] Nazarov, F. and Sodin, M. (2011). Fluctuations in random complex zeroes: Asymptotic normality revisited. Int. Math. Res. Not. IMRN 24 5720–5759.
  • [37] Nazarov, F. and Sodin, M. (2012). Correlation functions for random complex zeroes: Strong clustering and local universality. Comm. Math. Phys. 310 75–98.
  • [38] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192. Cambridge Univ. Press, Cambridge.
  • [39] Peccati, G. and Reitzner, M., eds. (2016). Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener–Itô Chaos Expansions and Stochastic Geometry. Bocconi & Springer Series 7. Springer, Cham.
  • [40] Penrose, M. D. (2007). Gaussian limits for random geometric measures. Electron. J. Probab. 12 989–1035.
  • [41] Penrose, M. D. (2007). Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13 1124–1150.
  • [42] Penrose, M. D. and Yukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 1005–1041.
  • [43] Penrose, M. D. and Yukich, J. E. (2002). Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12 272–301.
  • [44] Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 277–303.
  • [45] Penrose, M. D. and Yukich, J. E. (2005). Normal approximation in geometric probability. In Stein’s Method and Applications. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 5 37–58. Singapore Univ. Press, Singapore.
  • [46] Poinas, A., Delyon, B. and Lavancier, F. (2017). Mixing properties and central limit theorem for associated point processes. Preprint. Available at arXiv:1705.02276v2.
  • [47] Reddy, R. T., Vadlamani, S. and Yogeshwaran, D. (2018). Central limit theorem for exponentially quasi-local statistics of spin models on Cayley graphs. J. Stat. Phys. 173 941–984.
  • [48] Reitzner, M. and Schulte, M. (2013). Central limit theorems for $U$-statistics of Poisson point processes. Ann. Probab. 41 3879–3909.
  • [49] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
  • [50] Schreiber, T. and Yukich, J. E. (2013). Limit theorems for geometric functionals of Gibbs point processes. Ann. Inst. Henri Poincaré Probab. Stat. 49 1158–1182.
  • [51] Schulte, M. and Thäle, C. (2017). Central limit theorems for the radial spanning tree. Random Structures Algorithms 50 262–286.
  • [52] 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.
  • [53] Soshnikov, A. (2002). Gaussian limit for determinantal random point fields. Ann. Probab. 30 171–187.
  • [54] Xia, A. and Yukich, J. E. (2015). Normal approximation for statistics of Gibbsian input in geometric probability. Adv. in Appl. Probab. 47 934–972.
  • [55] Yogeshwaran, D. and Adler, R. J. (2015). On the topology of random complexes built over stationary point processes. Ann. Appl. Probab. 25 3338–3380.
  • [56] Yukich, J. (2013). Limit theorems in discrete stochastic geometry. In Stochastic Geometry, Spatial Statistics and Random Fields (E. Spodarev, ed.). Lecture Notes in Math. 2068 239–275. Springer, Heidelberg.
  • [57] Yukich, J. E. (1998). Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Math. 1675. Springer, Berlin.

Supplemental materials

  • Supplement to “Limit theory for geometric statistics of point processes having fast decay of correlations”. This supplement contains various auxiliary facts needed in the proofs. These facts, some of which are of independent interest, may also be found in the arXiv version [10] of this paper.