Covering One Point Process with Another

Let $$ and $$ be i.i.d. random uniform points in a bounded domain $$ with smooth or polygonal boundary. Given $$ , define the two-sample k-coverage threshold $$ to be the smallest r such that each point of $$ is covered at least k times by the disks of radius r centred on $$ . We obtain the limiting distribution of $$ as $$ with $$ for some constant $$ , with k fixed. If A has unit area, then $$ is asymptotically Gumbel distributed with scale parameter 1 and location parameter $$ . For $$ , we find that $$ is asymptotically Gumbel with scale parameter 2 and a more complicated location parameter involving the perimeter of A; boundary effects dominate when $$ . For $$ 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.