Moderate deviations for stabilizing functionals in geometric probability

TL;DR

This paper derives explicit bounds on moderate deviation probabilities for geometric functionals with stabilization properties under Poisson input, using cumulant expansions, and introduces a new criterion for non-degenerate variance.

Contribution

It provides a new approach to bounding deviations and a criterion for variance non-degeneracy, extending CLTs to broader settings without bounded support assumptions.

Findings

Explicit bounds on deviation probabilities for geometric functionals.

A new criterion for the limiting variance to be non-degenerate.

Application to models like random packing, nearest neighbors, and influence graphs.

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.