Covering One Point Process with Another

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 thresholdRn,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.