Moderate deviations for stabilizing functionals in geometric probability

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

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 89-128.

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.

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