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

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 be the expected volume of a ball of radius Y, and suppose r=r (t) is chosen so that t rᵈ - t - (d-1) 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 ( (t) / t) bound on the rate of convergence in that result. With a slight adjustment to r (t), this can be improved to O (1/ t).