Covering one point process with another

Abstract

Let $X_1,X_2,\dots$ and $Y_1,Y_2,\dots$ 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 $\{Y_1,\dots ,Y_m\}$ is covered at least $k$ times by the disks of radius $r$ centred on $X_1,\dots ,X_n$. We obtain the limiting distribution of $R_{n,m,k}$ as $n\to \infty$ with $m=m(n)\sim \ddot{I}„n$ for some constant $\ddot{I}„>0$, with $k$ fixed. If $A$ has unit area, then $n\ddot{I}€R_{n,m(n),1}^2-\log n$ is asymptotically Gumbel distributed with scale parameter $1$ and location parameter $\log \ddot{I}„$. For $k>2$, we find that $n\ddot{I}€R_{n,m(n),k}^2-\log n-(2k-3)\log \log n$ 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 $k$.

35 pages, 6 figures