Random Euclidean coverage from within

• 成果类型: Article
• 署名作者: Penrose, Mathew D.
• 署名单位: University of Bath
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01182-5
• 发表日期: 2023
• 页码: 747-814
• 关键词: maximal-spacings

摘要

Let X-1, X-2, . . . be independent random uniform points in a bounded domain A subset of R-d with smooth boundary. Define the coverage threshold Rn 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 |& part; A|, if d = 3 then P[n pi R-n(3) - log n - 2 log(log n) <= x] converges to exp(-2(-4)pi(5/3)|& part; A|e(-2x/3)) and (n pi R-n(3))/(log n) -> 1 almost surely, and if d = 2 then P[n pi R-n(2) - log n - log(log n) <= x] converges to exp(-e(-x )- |& part; A|pi(-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.