Papers › Covering one point process with another
Covering one point process with another
Frankie Higgs, Mathew D. Penrose, Xiaochuan Yang
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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 {\em two-sample k-coverage threshold} R_(n,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 R_(n,m,k) as n →∞ with m= m(n) ∼τn for some constant τ>0, with $k $ fixed. If A has unit area, then n πR_(n,m(n),1)² - logn is asymptotically Gumbel distributed with scale parameter $1$ and location parameter logτ. For k >2, we find that n πR_(n,m(n),k)² - 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.
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