Moderate deviations for stabilizing functionals in geometric probability
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 89-128.

The purpose of the present paper is to establish explicit upper and lower bounds on moderate deviation probabilities for a rather general class of geometric functionals enjoying the stabilization property, under Poisson input and the assumption of a certain control over the growth of the moments of the functional and its radius of stabilization. Our proof techniques rely on cumulant expansions and cluster measures. In addition, we establish a new criterion for the limiting variance to be non-degenerate. Moreover, our main result provides a new central limit theorem, which, though stated under strong moment assumptions, does not require bounded support of the intensity of the Poisson input. We apply our results to three groups of examples: random packing models, geometric functionals based on Euclidean nearest neighbors and the sphere of influence graphs.

L’objectif de cet article est d’établir une majoration et une minoration explicite pour les probabilités des déviations modérées d’une classe assez générale de fonctionnelles géométriques possédant une propriété de stabilisation pour des données de Poisson et sous l’hypothèse d’un contrôle de la croissance des moments de la fonctionnelle et de son rayon de stabilisation. Les techniques utilisées dans les preuves reposent sur des développements de cumulants et des mesures de clusters. En outre, nous proposons un nouveau critère pour que la variance limite soit non-dégénérée. De plus, notre résultat principal fournit un nouveau théorème central limite, qui, bien que formulé sous une hypothèse assez forte sur les moments, ne nécessite pas que l’intensité des données de Poisson ait un support borné. Nous appliquons nos résultats à trois groupes d’exemples: les modèles de pavages aléatoires, les fonctionnelles géométriques dépendantes des voisins les plus proches en distance euclidienne et les graphes des sphères d’influence.

DOI : 10.1214/13-AIHP576
Classification : 60F10, 60D05
Keywords: stabilizing functionals, moderate deviations, explicit bounds, cumulants, random packing, random graphs
@article{AIHPB_2015__51_1_89_0,
     author = {Eichelsbacher, P. and Rai\v{c}, M. and Schreiber, T.},
     title = {Moderate deviations for stabilizing functionals in geometric probability},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {89--128},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {1},
     year = {2015},
     doi = {10.1214/13-AIHP576},
     mrnumber = {3300965},
     zbl = {1312.60033},
     language = {en},
     url = {https://www.numdam.org/articles/10.1214/13-AIHP576/}
}
TY  - JOUR
AU  - Eichelsbacher, P.
AU  - Raič, M.
AU  - Schreiber, T.
TI  - Moderate deviations for stabilizing functionals in geometric probability
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2015
SP  - 89
EP  - 128
VL  - 51
IS  - 1
PB  - Gauthier-Villars
UR  - https://www.numdam.org/articles/10.1214/13-AIHP576/
DO  - 10.1214/13-AIHP576
LA  - en
ID  - AIHPB_2015__51_1_89_0
ER  - 
%0 Journal Article
%A Eichelsbacher, P.
%A Raič, M.
%A Schreiber, T.
%T Moderate deviations for stabilizing functionals in geometric probability
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2015
%P 89-128
%V 51
%N 1
%I Gauthier-Villars
%U https://www.numdam.org/articles/10.1214/13-AIHP576/
%R 10.1214/13-AIHP576
%G en
%F AIHPB_2015__51_1_89_0
Eichelsbacher, P.; Raič, M.; Schreiber, T. Moderate deviations for stabilizing functionals in geometric probability. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 89-128. doi : 10.1214/13-AIHP576. https://www.numdam.org/articles/10.1214/13-AIHP576/

[1] Y. Baryshnikov, P. Eichelsbacher, T. Schreiber and J. E. Yukich. Moderate deviations for some point measures in geometric probability. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 422–446. | DOI | Numdam | MR | Zbl

[2] Y. Baryshnikov and J. E. Yukich. Gaussian fields and random packing. J. Statist. Phys. 111 (2003) 443–463. | DOI | MR | Zbl

[3] Y. Baryshnikov and J. E. Yukich. Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 (2005) 213–253. | DOI | MR | Zbl

[4] V. Bentkus and R. Rudzkis. On exponential estimates of the distribution of random variables. Litovsk. Mat. Sb. 20 (1980) 15–30 (in Russian). | MR | Zbl

[5] T. Bodineau, D. Ioffe and Y. Velenik. Rigorous probabilistic analysis of equilibrium crystal shapes. J. Math. Phys. 41 (2000) 1033–1098. | DOI | MR | Zbl

[6] S. N. Chiu and M. P. Quine. Central limit theory for the number of seeds in a growth model in d with inhomogeneous Poisson arrivals. Ann. Appl. Probab. 7 (1997) 802–814. | MR | Zbl

[7] E. G. Coffman Jr., L. Flatto, P. Jelenković and B. Poonen. Packing random intervals on-line. Average-case analysis of algorithms. Algorithmica 22 (1998) 448–476. | MR | Zbl

[8] D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes, 2nd edition. Elementary Theory and Methods I. Springer, New York, 2003. | MR | Zbl

[9] D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes, 2nd edition. General Theory and Structure II. Springer, New York, 2008. | MR | Zbl

[10] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998. | DOI | MR | Zbl

[11] R. Dobrushin, R. Kotecký and S. Shlosman. Wulff Construction. A Global Shape from Local Interaction. American Mathematical Society, Providence, RI, 1992. | MR | Zbl

[12] A. Dvoretzky and H. Robbins. On the “parking” problem. Magyar Tud. Akad. Mat. Kutató Int. Közl. 9 (1964) 209–225. | MR | Zbl

[13] P. Eichelsbacher and T. Schreiber. Process level moderate deviations for stabilizing functionals. ESAIM Probab. Stat. 14 (2010) 1–15. | DOI | Numdam | MR | Zbl

[14] A. B. Gorchakov. Upper bounds for cumulants of the sum of multi-indexed random variables. Discrete Math. Appl. 5 (1995) 317–331. | DOI | MR | Zbl

[15] M. Hardy. Combinatorics of partial derivatives. Electron. J. Combin. 13 (2006) Research Paper 1 (electronic). | MR | Zbl

[16] L. Heinrich. Large deviations of the empirical volume fraction for stationary Poisson grain models. Ann. Appl. Probab. 15 (2005) 392–420. | DOI | MR | Zbl

[17] L. Heinrich and M. Spiess. Berry–Esséen bounds and Cramér-type large deviations for the volume distribution of Poisson cylinder processes. Lith. Math. J. 49 (2009) 381–398. | DOI | MR | Zbl

[18] K. Krickeberg. Moments of point processes. In Probability and Information Theory, II 70–101. Springer, Berlin, 1973. | MR | Zbl

[19] V. A. Malyshev and R. A. Minlos. Gibbs Random Fields. Kluwer Academic, Dordrecht, 1991. | DOI | MR | Zbl

[20] M. D. Penrose. Random parking, sequential adsorption, and the jamming limit. Comm. Math. Phys. 218 (2001) 153–176. | DOI | MR | Zbl

[21] M. D. Penrose. Random Geometric Graphs. Oxford Univ. Press, Oxford, 2003. | DOI | MR | Zbl

[22] M. D. Penrose. Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Probab. 33 (2005) 1945–1991. | DOI | MR | Zbl

[23] M. D. Penrose. Convergence of random measures in geometric probability. Preprint, 2005. Available at arXiv:math/0508464.

[24] M. D. Penrose. Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13 (2007) 1124–1150. | DOI | MR | Zbl

[25] M. D. Penrose. Gaussian limits for random geometric measures. Electron. J. Probab. 12 (2007) 989–1035 (electronic). | DOI | MR | Zbl

[26] M. D. Penrose and A. R. Wade. Multivariate normal approximation in geometric probability. J. Stat. Theory Pract. 2 (2008) 293–326. | DOI | MR | Zbl

[27] M. D. Penrose and J. E. Yukich. Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 (2001) 1005–1041. | DOI | MR | Zbl

[28] M. D. Penrose and J. E. Yukich. Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12 (2002) 272–301. | DOI | MR | Zbl

[29] M. D. Penrose and J. E. Yukich. Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 (2003) 277–303. | DOI | MR | Zbl

[30] M. D. Penrose and J. E. Yukich. Normal approximation in geometric probability. In Stein’s Method and Applications 37–58. Singapore Univ. Press, Singapore, 2005. | MR

[31] A. Rényi. Théorie des éléments saillants d’une suite d’observations. Ann. Fac. Sci. Univ. Clermont-Ferrand 8 (1962) 7–13. | Numdam | MR | Zbl

[32] R. Rudzkis L. Saulis and V. Statulevičius. A general lemma on probabilities of large deviations. Lith. Math. J. 18 (1978) 226–238. | MR | Zbl

[33] L. Saulis and V. Statulevičius. Limit Theorems on Large Deviations. Kluwer Academic, Dordrecht, 1991. | DOI | MR | Zbl

[34] T. Schreiber and J. E. Yukich. Large deviations for functionals of spatial point processes with applications to random packing and spatial graphs. Stochastic Process. Appl. 115 (2005) 1332–1356. | DOI | MR | Zbl

  • Basrak, Bojan; Molchanov, Ilya; Planinić, Hrvoje Extremal behavior of stationary marked point processes, Advances in Applied Probability (2025), p. 1 | DOI:10.1017/apr.2025.23
  • Heerten, Nils; Hirsch, Christian; Otto, Moritz Moderate deviations for weight-dependent random connection models, Journal of Applied Probability (2025), p. 1 | DOI:10.1017/jpr.2025.13
  • Bhattacharjee, Chinmoy; Molchanov, Ilya; Turin, Riccardo Central limit theorem for a birth–growth model with poisson arrivals and random growth speed, Advances in Applied Probability, Volume 56 (2024) no. 3, p. 1004 | DOI:10.1017/apr.2023.58
  • Botnan, Magnus B.; Hirsch, Christian On the consistency and asymptotic normality of multiparameter persistent Betti numbers, Journal of Applied and Computational Topology, Volume 8 (2024) no. 6, p. 1465 | DOI:10.1007/s41468-022-00110-9
  • Cipriani, Alessandra; Hirsch, Christian; Vittorietti, Martina Topology-based goodness-of-fit tests for sliced spatial data, Computational Statistics Data Analysis, Volume 179 (2023), p. 107655 | DOI:10.1016/j.csda.2022.107655
  • Bonnet, Gilles; Hirsch, Christian; Rosen, Daniel; Willhalm, Daniel Limit theory of sparse random geometric graphs in high dimensions, Stochastic Processes and their Applications, Volume 163 (2023), p. 203 | DOI:10.1016/j.spa.2023.06.002
  • Biscio, Christophe A. N.; Svane, Anne Marie A functional central limit theorem for the empirical Ripley’s K-function, Electronic Journal of Statistics, Volume 16 (2022) no. 1 | DOI:10.1214/22-ejs2017
  • Döring, Hanna; Jansen, Sabine; Schubert, Kristina The method of cumulants for the normal approximation, Probability Surveys, Volume 19 (2022) no. none | DOI:10.1214/22-ps7
  • Gusakova, Anna; Thäle, Christoph The volume of simplices in high-dimensional Poisson–Delaunay tessellations, Annales Henri Lebesgue, Volume 4 (2021), p. 121 | DOI:10.5802/ahl.68
  • Eichelsbacher, Peter; Knichel, Lukas Fine asymptotics for models with Gamma type moments, Random Matrices: Theory and Applications, Volume 10 (2021) no. 01, p. 2150007 | DOI:10.1142/s2010326321500076
  • Hirsch, Christian; Jahnel, Benedikt; Tóbiás, András Lower large deviations for geometric functionals, Electronic Communications in Probability, Volume 25 (2020) no. none | DOI:10.1214/20-ecp322
  • Biscio, Christophe A. N.; Chenavier, Nicolas; Hirsch, Christian; Svane, Anne Marie Testing goodness of fit for point processes via topological data analysis, Electronic Journal of Statistics, Volume 14 (2020) no. 1 | DOI:10.1214/20-ejs1683
  • Lachièze-Rey, Raphaël; Schulte, Matthias; Yukich, J. E. Normal approximation for stabilizing functionals, The Annals of Applied Probability, Volume 29 (2019) no. 2 | DOI:10.1214/18-aap1405
  • Błaszczyszyn, B.; Yogeshwaran, D.; Yukich, J. E. Limit theory for geometric statistics of point processes having fast decay of correlations, The Annals of Probability, Volume 47 (2019) no. 2 | DOI:10.1214/18-aop1273
  • Grote, Julian; Thäle, Christoph Concentration and moderate deviations for Poisson polytopes and polyhedra, Bernoulli, Volume 24 (2018) no. 4A | DOI:10.3150/17-bej946
  • Grote, Julian; Thäle, Christoph Gaussian polytopes: A cumulant-based approach, Journal of Complexity, Volume 47 (2018), p. 1 | DOI:10.1016/j.jco.2018.03.001
  • Dhara, Souvik; van Leeuwaarden, Johan S. H.; Mukherjee, Debankur Corrected Mean-Field Model for Random Sequential Adsorption on Random Geometric Graphs, Journal of Statistical Physics, Volume 173 (2018) no. 3-4, p. 872 | DOI:10.1007/s10955-018-2019-8
  • Bachmann, Sascha; Reitzner, Matthias Concentration for Poisson U-statistics: Subgraph counts in random geometric graphs, Stochastic Processes and their Applications, Volume 128 (2018) no. 10, p. 3327 | DOI:10.1016/j.spa.2017.11.001
  • Reitzner, Matthias; Schulte, Matthias; Thäle, Christoph Limit theory for the Gilbert graph, Advances in Applied Mathematics, Volume 88 (2017), p. 26 | DOI:10.1016/j.aam.2016.12.006
  • Bachmann, Sascha; Peccati, Giovanni Concentration bounds for geometric Poisson functionals: Logarithmic Sobolev inequalities revisited, Electronic Journal of Probability, Volume 21 (2016) no. none | DOI:10.1214/16-ejp4235
  • Schulte, Matthias; Thäle, Christoph Cumulants on Wiener chaos: Moderate deviations and the fourth moment theorem, Journal of Functional Analysis, Volume 270 (2016) no. 6, p. 2223 | DOI:10.1016/j.jfa.2016.01.002
  • Xia, Aihua; Yukich, J. E. Normal approximation for statistics of Gibbsian input in geometric probability, Advances in Applied Probability, Volume 47 (2015) no. 04, p. 934 | DOI:10.1017/s0001867800048965
  • Xia, Aihua; Yukich, J. E. Normal approximation for statistics of Gibbsian input in geometric probability, Advances in Applied Probability, Volume 47 (2015) no. 4, p. 934 | DOI:10.1239/aap/1449859795
  • Reitzner, Matthias Poisson point processes: large deviation inequalities for the convex distance, Electronic Communications in Probability, Volume 18 (2013) no. none | DOI:10.1214/ecp.v18-2851

Cité par 24 documents. Sources : Crossref