On the rate of convergence in the Hall-Janson coverage theorem

This is the authors' abstract. We don't add key points for this paper.

Consider a spherical Poisson Boolean model Z in Euclidean d-space with d ≥ 2, with Poisson intensity t and radii distributed like $rY$ with r ≥ 0 a scaling parameter and Y a fixed nonnegative random variable with finite $(2d)$-th moment. Let A ⊂ Rᵈ be compact with a nice boundary. Let α be the expected volume of a ball of radius Y, and suppose $r=r(t)$ is chosen so that α t rᵈ - log t - (d-1) log log t is a constant indepenent of t. A classical result of Hall and of Janson determines the (non-trivial) large-t limit of the probability that A is fully covered by Z. In this paper we provide an O((log log t)/log t) bound on the rate of convergence in that result. With a slight adjustment to $r(t)$, this can be improved to O(1/log t).