Refined convergence for the Boolean model

Pierre Calka (1) , Julien Michel (2) , Katy Paroux (3, 4)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
3 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
4 DIONYSOS - Dependability Interoperability and perfOrmance aNalYsiS Of networkS

In a previous work, two of the authors proposed a new proof of a well known convergence result for the scaled elementary connected vacant component in the high intensity Boolean model towards the Crofton cell of the Poisson hyperplane process. In this paper, we consider the particular case of the two-dimensional Boolean model where the grains are discs with random radii. We investigate the second-order term in this convergence when the Boolean model and the Poisson line process are coupled on the same probability space. A precise coupling between the Boolean model and the Poisson line process is first established, a result of directional convergence in distribution for the difference of the two sets involved is derived as well.

Keywords: Poisson point process; Crofton cell; convergence; stochastic geometry

Domains: Probability [math.PR]

Pierre Calka, Julien Michel, Katy Paroux. Refined convergence for the Boolean model. Advances in Applied Probability, 2009, 41 (4), pp.940-957. ⟨hal-00274553v2⟩