Covering One Point Process with Another

Let $$X_1,X_2, \ldots $$ X 1 , X 2 , … and $$Y_1, Y_2, \ldots $$ Y 1 , Y 2 , … be i.i.d. random uniform points in a bounded domain $$A \subset \mathbb {R}^2$$ A ⊂ R 2 with smooth or polygonal boundary. Given $$n,m,k \in \mathbb {N}$$ n , m , k ∈ N , define the two-sample k-coverage threshold $$R_{n,m,k}$$ R n , m , k to be the smallest r such that each point of $$ \{Y_1,\ldots ,Y_m\}$$ { Y 1 , … , Y m } is covered at least k times by the disks of radius r centred on $$X_1,\ldots ,X_n$$ X 1 , … , X n . We obtain the limiting distribution of $$R_{n,m,k}$$ R n , m , k as $$n \rightarrow \infty $$ n → ∞ with $$m= m(n) \sim \tau n$$ m = m ( n ) ∼ τ n for some constant $$\tau >0$$ τ > 0 , with k fixed. If A has unit area, then $$n \pi R_{n,m(n),1}^2 - \log n$$ n π R n , m ( n ) , 1 2 - log n is asymptotically Gumbel distributed with scale parameter 1 and location parameter $$\log \tau $$ log τ . For $$k >2$$ k > 2 , we find that $$n \pi R_{n,m(n),k}^2 - \log n - (2k-3) \log \log n$$ n π R n , m ( n ) , k 2 - log n - ( 2 k - 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 $$k >2$$ k > 2 . For $$k=2$$ 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.