Moderate deviations for stabilizing functionals in geometric probability

Abstract

The purpose of the present paper is to establish explicit 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 and yield completely explicit bounds on deviation probabilities. 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.

52 pages

References in corpus (6)

• Gaussian limits for random measures in geometric probability
• Laws of large numbers in stochastic geometry with statistical applications
• Large deviations of the empirical volume fraction for stationary Poisson grain models
• Moderate deviations for some point measures in geometric probability
• Multivariate normal approximation in geometric probability
• Convergence of random measures in geometric probability

Cited by in corpus (5)

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• Corrected mean-field model for random sequential adsorption on random geometric graphs
• Limit theory of sparse random geometric graphs in high dimensions
• The volume of simplices in high-dimensional Poisson-Delaunay tessellations
• Extremal behavior of stationary marked point processes