Volume 5 (2022)
Bobrowski, Omer; Schulte, Matthias; Yogeshwaran, D.
Poisson process approximation under stabilization and Palm coupling
Annales Henri Lebesgue, Volume 5 (2022), pp. 1489-1534.

Metadata

DOI10.5802/ahl.156
Keywords Functional limit theorems, Poisson process approximation, Kantorovich-Rubinstein distance, Point processes, Stein’s method, Glauber dynamics, Palm coupling, Stabilizing statistics, $k$-nearest neighbor balls, Morse critical points, Binomial point processes

Abstract

We present new Poisson process approximation results for stabilizing functionals of Poisson and binomial point processes. These functionals are allowed to have an unbounded range of interaction and encompass many examples in stochastic geometry. Our bounds are derived for the Kantorovich–Rubinstein distance using the generator approach to Stein’s method. We give different types of bounds for different point processes. While some of our bounds are given in terms of coupling of the point process with its Palm version, the others are in terms of the local dependence structure formalized via the notion of stabilization. We provide two supporting examples for our new framework – one is for Morse critical points of the distance function, and the other is for large k-nearest neighbor balls. Our bounds considerably extend the results in Barbour and Brown (1992), Decreusefond, Schulte and Thäle (2016) and Otto (2020).


References

[AGG89] Arratia, Richard A.; Goldstein, Larry; Gordon, Louis Two moments suffice for Poisson approximations: the Chen–Stein method, Ann. Probab., Volume 17 (1989) no. 1, pp. 9-25 | MR | Zbl

[BA14] Bobrowski, Omer; Adler, Robert J. Distance functions, critical points, and the topology of random Čech complexes, Homology Homotopy Appl., Volume 16 (2014) no. 2, pp. 311-344 | DOI | Zbl

[Bar88] Barbour, Andrew D. Stein’s method and Poisson process convergence, J. Appl. Probab., Volume 25 (1988) no. A, pp. 175-184 (“A celebration of applied probability”, Spec. Vol.) | DOI | MR | Zbl

[BB92] Barbour, Andrew D.; Brown, Timothy C. Stein’s method and point process approximation, Stochastic Processes Appl., Volume 43 (1992) no. 1, pp. 9-31 | DOI | MR | Zbl

[BBK20] Baccelli, François; Blaszczyszyn, Bartłomiej; Karray, Mohamed Random Measures, Point Processes and Stochastic Geometry (2020) (https://hal.inria.fr/hal-02460214/)

[BHJ92] Barbour, Andrew D.; Holst, Lars; Janson, Svante Poisson approximation, Oxford Studies in Probability, 2, Oxford University Press, 1992 | Zbl

[BK18] Bobrowski, Omer; Kahle, Matthew Topology of random geometric complexes: a survey, J. Appl. Comput. Topol., Volume 1 (2018) no. 3-4, pp. 331-364 | DOI | MR | Zbl

[BL09] Baumstark, Volker; Last, Günter Gamma distributions for stationary Poisson flat processes, Adv. Appl. Probab., Volume 41 (2009) no. 4, pp. 911-939 | DOI | MR | Zbl

[BM22] Bhattacharjee, Chinmoy; Molchanov, Ilya Gaussian approximation for sums of region-stabilizing scores, Electron. J. Probab., Volume 27 (2022), 111 | DOI | MR | Zbl

[Bob22] Bobrowski, Omer Homological Connectivity in Random Čech Complexes, Probab. Theory Relat. Fields, Volume 183 (2022) no. 3-4, pp. 715-788 | DOI | MR | Zbl

[BW17] Bobrowski, Omer; Weinberger, Shmuel On the vanishing of homology in random Čech complexes, Random Struct. Algorithms, Volume 51 (2017) no. 1, pp. 14-51 | DOI | Zbl

[CC14] Calka, Pierre; Chenavier, Nicolas Extreme values for characteristic radii of a Poisson–Voronoi tessellation, Extremes, Volume 17 (2014) no. 3, pp. 359-385 | DOI | MR | Zbl

[Che14] Chenavier, Nicolas A general study of extremes of stationary tessellations with examples, Stochastic Processes Appl., Volume 124 (2014) no. 9, pp. 2917-2953 | DOI | MR | Zbl

[CHO21] Chenavier, Nicolas; Henze, Norbert; Otto, Moritz Limit laws for large kth-nearest neighbor balls (2021) (https://arxiv.org/abs/2105.00038)

[CX04] Chen, Louis H.; Xia, Aihua Stein’s method, Palm theory and Poisson process approximation, Ann. Probab., Volume 32 (2004) no. 3B, pp. 2545-2569 | MR | Zbl

[DST16] Decreusefond, Laurent; Schulte, Matthias; Thäle, Christoph Functional Poisson approximation in Kantorovich–Rubinstein distance with applications to U-statistics and stochastic geometry, Ann. Probab., Volume 44 (2016) no. 3, pp. 2147-2197 | MR | Zbl

[DVJ07] Daley, Daryl J.; Vere-Jones, David An introduction to the theory of point processes: volume II: general theory and structure, Probability and Its Applications, Springer, 2007 | Zbl

[ERS15] Eichelsbacher, Peter; Raič, Martin; Schreiber, Tomacz Moderate deviations for stabilizing functionals in geometric probability, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 51 (2015) no. 1, pp. 89-128 | Numdam | MR | Zbl

[GHW19] Györfi, Lázló; Henze, Norbert; Walk, Harro The limit distribution of the maximum probability nearest-neighbour ball, J. Appl. Probab., Volume 56 (2019) no. 2, pp. 574-589 | DOI | MR | Zbl

[GR97] Gershkovich, V. Ya.; Rubinsten, J. Hyam Morse theory for min-type functions, Asian J. Math., Volume 1 (1997) no. 4, pp. 696-715 | DOI | MR | Zbl

[Hat02] Hatcher, Allen E. Algebraic topology, Cambridge University Press, 2002 | Zbl

[IY20] Iyer, Srikanth K.; Yogeshwaran, D. Thresholds for vanishing of ‘Isolated’ faces in random Čech and Vietoris–Rips complexes, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 56 (2020) no. 3, pp. 1869-1897 | Zbl

[Kah11] Kahle, Matthew Random geometric complexes, Discrete Comput. Geom., Volume 45 (2011) no. 3, pp. 553-573 | DOI | MR | Zbl

[Kal02] Kallenberg, Olav Foundations of modern probability, Probability and Its Applications, Springer, 2002 | DOI | Zbl

[LP17] Last, Günter; Penrose, Mathew Lectures on the Poisson Process, Institute of Mathematical Statistics Textbooks, 7, Cambridge University Press, 2017 | DOI | Zbl

[LPY21] Last, Günter; Peccati, Giovanni; Yogeshwaran, D. Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees (2021) (https://arxiv.org/abs/2101.07180)

[LRSY19] Lachièze-Rey, Raphaël; Schulte, Matthias; Yukich, Joseph E. Normal approximation for stabilizing functionals, Ann. Appl. Probab., Volume 29 (2019) no. 2, pp. 931-993 | MR | Zbl

[Mil63] Milnor, John W. Morse theory. Based on lecture notes by M. Spivak and R. Wells, Annals of Mathematics Studies, 51, Princeton University Press, 1963 | Zbl

[OA17] Owada, Takhashi; Adler, Robert J. Limit theorems for point processes under geometric constraints (and topological crackle), Ann. Probab., Volume 45 (2017) no. 3, pp. 2004-2055 | MR | Zbl

[Ott20] Otto, Moritz Poisson approximation of Poisson-driven point processes and extreme values in stochastic geometry (2020) (https://arxiv.org/abs/2005.10116)

[Ott21] Otto, Moritz Extremal behavior of large cells in the Poisson hyperplane mosaic (2021) (https://arxiv.org/abs/2106.14823)

[Pen97] Penrose, Matthew D. The longest edge of the random minimal spanning tree, Ann. Appl. Probab., Volume 7 (1997) no. 2, pp. 340-361 | MR | Zbl

[Pen07] Penrose, Matthew D. Laws of large numbers in stochastic geometry with statistical applications, Bernoulli, Volume 13 (2007) no. 4, pp. 1124-1150 | MR | Zbl

[Pen18] Penrose, Matthew D. Inhomogeneous random graphs, isolated vertices, and Poisson approximation, J. Appl. Probab., Volume 55 (2018) no. 1, pp. 112-136 | DOI | MR | Zbl

[PG10] Penrose, Matthew D.; Goldstein, Larry Normal approximation for coverage processes over binomial point processes, Ann. Appl. Probab., Volume 20 (2010) no. 2, pp. 696-721 | Zbl

[Pre75] Preston, Chris Spatial birth and death processes, Adv. Appl. Probab., Volume 7 (1975) no. 3, pp. 465-466 | DOI | Zbl

[PS22] Pianoforte, Federico; Schulte, Matthias Criteria for Poisson process convergence with applications to inhomogeneous Poisson–Voronoi tessellations, Stochastic Processes Appl., Volume 147 (2022), pp. 388-422 | DOI | MR | Zbl

[Ros11] Ross, Nathan Fundamentals of Stein’s method, Probab. Surv., Volume 8 (2011), pp. 210-293 | MR | Zbl

[Sch05] Schuhmacher, Dominic Distance estimates for dependent superpositions of point processes, Stochastic Processes Appl., Volume 115 (2005) no. 11, pp. 1819-1837 | DOI | MR | Zbl

[Sch09] Schuhmacher, Dominic Stein’s method and Poisson process approximation for a class of Wasserstein metrics, Bernoulli, Volume 15 (2009) no. 2, pp. 550-568 | MR | Zbl

[Sch10] Schreiber, Tomacz Limit theorems in stochastic geometry, New perspectives in stochastic geometry (Kendall, Wilfrid S.; Molchanov, I., eds.), Oxford University Press, 2010, pp. 111-144 | Zbl

[Xia05] Xia, Aihua Stein’s method and Poisson process approximation, An introduction to Stein’s method (Barbour, A. D.; Chen, L. H. Y., eds.) (Lecture Notes Series. Institute for Mathematical Sciences), Volume 4, Singapore University Press, Singapore, 2005, pp. 115-181 | DOI | MR

[Yuk13] Yukich, Joseph E. Limit theorems in discrete stochastic geometry, Stochastic geometry, spatial statistics and random fields. Asymptotic methods (Spodarev, Evgeny, ed.) (Lecture Notes in Mathematics), Volume 2068, Springer, 2013, pp. 239-275 | DOI | MR | Zbl