Covering one point process with another

Abstract

Let $X_1,X_2, \ldots $ and $Y_1, Y_2, \ldots$ be i.i.d. random uniform points in a bounded domain $A \subset \mathbb{R}^2$ with smooth or polygonal boundary. Given $n,m,k \in \mathbb{N}$, define the {\em two-sample $k$-coverage threshold} $R_{n,m,k}$ to be the smallest $r$ such that each point of $ \{Y_1,\ldots,Y_m\}$ is covered at least $k$ times by the disks of radius $r$ centred on $X_1,\ldots,X_n$. We obtain the limiting distribution of $R_{n,m,k}$ as $n \to \infty$ with $m= m(n) \sim \tau n$ for some constant $\tau >0$, with $k $ fixed. If $A$ has unit area, then $n \pi R_{n,m(n),1}^2 - \log n$ is asymptotically Gumbel distributed with scale parameter $1$ and location parameter $\log \tau$. For $k >2$, we find that $n \pi R_{n,m(n),k}^2 - \log n - (2k-3) \log \log n$ is asymptotically Gumbel with scale parameter $2$ and a more complicated location parameter involving the perimeter of $A$; boundary effects dominate when $k >2$. For $k=2$ the limiting cdf is a two-component extreme value distribution with scale parameters 1 and 2. We also give analogous results for higher dimensions, where the boundary effects dominate for all $k$.