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-2)-nd moment (or if d=2, a finite (2 + ε)-moment condition for some ε >0). Let A ⊂ R^d 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 r^d - log t - (d-1) loglog t is a constant independent 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((loglog 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).