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

TL;DR

This paper establishes the rate at which the probability of full coverage in a spherical Poisson Boolean model converges to its limit as the intensity grows, improving understanding of coverage thresholds in high-dimensional spaces.

Contribution

It provides explicit bounds on the convergence rate in the Hall-Janson coverage theorem, refining previous asymptotic results with quantitative estimates.

Findings

Rate of convergence is $O(( ext{log log } t)/ ext{log } t)$

Improved rate to $O(1/ ext{log } t$ with slight adjustment

Quantitative bounds enhance understanding of coverage probabilities

Abstract

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 $O(1/\log t)$.