Covering One Point Process with Another

• Published in: Methodology and computing in applied probability Vol. 27; no. 2; p. 40
• Main Authors: Higgs, Frankie, Penrose, Mathew D., Yang, Xiaochuan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2025; Springer Nature B.V
• Subjects: Anniversaries; Business and Management; Connectivity; Constraining; Economics; Electrical Engineering; Extreme values; Life Sciences; Mathematics and Statistics; Parameters; Random variables; Sensors; Statistics; 60D05; Weak limit; Poisson point process; Coverage threshold; 60F05; 60F15
• ISSN: 1387-5841; 1573-7713
• DOI: 10.1007/s11009-025-10165-7

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