Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Moderate deviations for stabilizing functionals in geometric probability

P. Eichelsbacher, M. Raič, and T. Schreiber

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 1 (2015), 89-128.

Dates
First available in Project Euclid: 14 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1421244400

Digital Object Identifier
doi:10.1214/13-AIHP576

Mathematical Reviews number (MathSciNet)
MR3300965

Zentralblatt MATH identifier
1312.60033

Subjects
Primary: 60F10: Large deviations 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Stabilizing functionals Moderate deviations Explicit bounds Cumulants Random packing Random graphs

Citation

Eichelsbacher, P.; Raič, M.; Schreiber, T. Moderate deviations for stabilizing functionals in geometric probability. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 1, 89--128. doi:10.1214/13-AIHP576. https://projecteuclid.org/euclid.aihp/1421244400


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