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 $A\subset R^2$ with smooth or polygonal boundary. Given $n,m,k\in N$ , define the 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 $m=m\left(n\right)\sim \tau n$ for some constant $\tau >0$ , with k fixed. If A has unit area, then $n\pi R_{n,m\left(n\right),1}^2-logn$ is asymptotically Gumbel distributed with scale parameter 1 and location parameter $log\tau$ . For $k>2$ , we find that $n\pi R_{n,m\left(n\right),k}^2-logn-(2k-3)loglogn$ 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.