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Abstract
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 largen limit. For example, if A has volume 1 and perimeter  ∂A , if d= 3 then P[nπRn3logn2log(logn)≤x] converges to exp ( 2 ^{ 4}π ^{5 / 3} ∂A e ^{} ^{2} ^{x} ^{/} ^{3}) and (nπRn3)/(logn)→1 almost surely, and if d= 2 then P[nπRn2lognlog(logn)≤x] converges to exp ( e ^{} ^{x}  ∂A π ^{ 1 / 2}e ^{} ^{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 language  English 

Pages (fromto)  747814 
Number of pages  68 
Journal  Probability Theory and Related Fields 
Volume  185 
Issue number  34 
Early online date  29 Jan 2023 
DOIs  
Publication status  Published  Apr 2023 
Bibliographical note
FundRef Engineering and Physical Sciences Research CouncilGrant number EP/T028653/1
Keywords
 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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Coverage and connectivity in stochastic geometry
Engineering and Physical Sciences Research Council
15/12/20 → 15/03/25
Project: Research council