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

Consider a spherical Poisson Boolean model $Z$ in Euclidean $d$-space with $d \geq 2$, with Poisson intensity $t$ and radii distributed like $rY$ with $r \geq 0$ a scaling parameter and $Y$ a fixed nonnegative random variable with finite $(2d-2)$-nd moment (or if $d=2$, a finite $(2 + \varepsilon)$-moment condition for some $\varepsilon >0$). Let $A \subset {\bf R}^d$ be compact with a nice boundary. Let $\alpha$ be the expected volume of a ball of radius $Y$, and suppose $r=r(t)$ is chosen so that $\alpha t r^d - \log t - (d-1) \log \log t$ 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 $O((\log \log t)/\log t)$ bound on the rate of convergence in that result. With a slight adjustment to $r(t)$, this can be improved to $O(1/\log t)$.