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 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 \(n \rightarrow \infty \) 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.
