Abstract
Let X1, X2, … and Y1, Y2, … be i.i.d. random uniform points in a bounded domain A ⊂ R2 with smooth or polygonal boundary. Given n, m, k ∈ N , define the two-sample k-coverage threshold Rn,m,k to be the smallest r such that each point of {Y1, …, Ym} is covered at least k times by the disks of radius r centred on X1, …, Xn . We obtain the limiting distribution of Rn,m,k as n → ∞ with m = m(n) ∼ τn for some constant τ > 0 , with k fixed. If A has unit area, then nπRn,m(n),12 - logn is asymptotically Gumbel distributed with scale parameter 1 and location parameter logτ . For k > 2 , we find that nπRn,m(n),k2 - 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.