Let X 1, X 2, … be independent random uniform points in a bounded domain A⊂ R d with smooth boundary. Define the coverage thresholdR n to be the smallest r such that A is covered by the balls of radius r centred on X 1, … , X n. We obtain the limiting distribution of R n and also a strong law of large numbers for R n in the large-n limit. For example, if A has volume 1 and perimeter | ∂A| , if d= 3 then P[nπRn3-logn-2log(logn)≤x] converges to exp (- 2 - 4π 5 / 3| ∂A| e - 2 x / 3) and (nπRn3)/(logn)→1 almost surely, and if d= 2 then P[nπRn2-logn-log(logn)≤x] converges to exp (- e - x- | ∂A| π - 1 / 2e - x / 2). We give similar results for general d, and also for the case where A is a polytope. We also generalize to allow for multiple coverage. The analysis relies on classical results by Hall and by Janson, along with a careful treatment of boundary effects. For the strong laws of large numbers, we can relax the requirement that the underlying density on A be uniform.

Original languageEnglish
Pages (from-to)747-814
Number of pages68
JournalProbability Theory and Related Fields
Issue number3-4
Early online date29 Jan 2023
Publication statusPublished - Apr 2023

Bibliographical note

FundRef Engineering and Physical Sciences Research Council
Grant number EP/T028653/1


  • Boolean model
  • Coverage threshold
  • Poisson point process
  • Strong law of large numbers
  • Weak limit

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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