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.
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.
Mots clés : 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 = {http://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 - http://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 http://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. http://www.numdam.org/articles/10.1214/13-AIHP576/
[1] Moderate deviations for some point measures in geometric probability. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 422–446. | DOI | Numdam | MR | Zbl
, , and .[2] Gaussian fields and random packing. J. Statist. Phys. 111 (2003) 443–463. | DOI | MR | Zbl
and .[3] Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 (2005) 213–253. | DOI | MR | Zbl
and .[4] On exponential estimates of the distribution of random variables. Litovsk. Mat. Sb. 20 (1980) 15–30 (in Russian). | MR | Zbl
and .[5] Rigorous probabilistic analysis of equilibrium crystal shapes. J. Math. Phys. 41 (2000) 1033–1098. | DOI | MR | Zbl
, and .[6] Central limit theory for the number of seeds in a growth model in with inhomogeneous Poisson arrivals. Ann. Appl. Probab. 7 (1997) 802–814. | MR | Zbl
and .[7] Packing random intervals on-line. Average-case analysis of algorithms. Algorithmica 22 (1998) 448–476. | MR | Zbl
, , and .[8] An Introduction to the Theory of Point Processes, 2nd edition. Elementary Theory and Methods I. Springer, New York, 2003. | MR | Zbl
and .[9] An Introduction to the Theory of Point Processes, 2nd edition. General Theory and Structure II. Springer, New York, 2008. | MR | Zbl
and .[10] Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998. | DOI | MR | Zbl
and .[11] Wulff Construction. A Global Shape from Local Interaction. American Mathematical Society, Providence, RI, 1992. | MR | Zbl
, and .[12] On the “parking” problem. Magyar Tud. Akad. Mat. Kutató Int. Közl. 9 (1964) 209–225. | MR | Zbl
and .[13] Process level moderate deviations for stabilizing functionals. ESAIM Probab. Stat. 14 (2010) 1–15. | DOI | Numdam | MR | Zbl
and .[14] Upper bounds for cumulants of the sum of multi-indexed random variables. Discrete Math. Appl. 5 (1995) 317–331. | DOI | MR | Zbl
.[15] Combinatorics of partial derivatives. Electron. J. Combin. 13 (2006) Research Paper 1 (electronic). | MR | Zbl
.[16] Large deviations of the empirical volume fraction for stationary Poisson grain models. Ann. Appl. Probab. 15 (2005) 392–420. | DOI | MR | Zbl
.[17] 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
and .[18] Moments of point processes. In Probability and Information Theory, II 70–101. Springer, Berlin, 1973. | MR | Zbl
.[19] Gibbs Random Fields. Kluwer Academic, Dordrecht, 1991. | DOI | MR | Zbl
and .[20] Random parking, sequential adsorption, and the jamming limit. Comm. Math. Phys. 218 (2001) 153–176. | DOI | MR | Zbl
.[21] Random Geometric Graphs. Oxford Univ. Press, Oxford, 2003. | DOI | MR | Zbl
.[22] Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Probab. 33 (2005) 1945–1991. | DOI | MR | Zbl
.[23] Convergence of random measures in geometric probability. Preprint, 2005. Available at arXiv:math/0508464.
.[24] Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13 (2007) 1124–1150. | DOI | MR | Zbl
.[25] Gaussian limits for random geometric measures. Electron. J. Probab. 12 (2007) 989–1035 (electronic). | DOI | MR | Zbl
.[26] Multivariate normal approximation in geometric probability. J. Stat. Theory Pract. 2 (2008) 293–326. | DOI | MR | Zbl
and .[27] Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 (2001) 1005–1041. | DOI | MR | Zbl
and .[28] Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12 (2002) 272–301. | DOI | MR | Zbl
and .[29] Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 (2003) 277–303. | DOI | MR | Zbl
and .[30] Normal approximation in geometric probability. In Stein’s Method and Applications 37–58. Singapore Univ. Press, Singapore, 2005. | MR
and .[31] Théorie des éléments saillants d’une suite d’observations. Ann. Fac. Sci. Univ. Clermont-Ferrand 8 (1962) 7–13. | Numdam | MR | Zbl
.[32] L. Saulis and V. Statulevičius. A general lemma on probabilities of large deviations. Lith. Math. J. 18 (1978) 226–238. | MR | Zbl
[33] Limit Theorems on Large Deviations. Kluwer Academic, Dordrecht, 1991. | DOI | MR | Zbl
and .[34] 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
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