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PreprintCC BY 4.0arXiv

Record times for coverage thresholds and maximal spacings

  • Mathew D. Penrose
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Abstract

Let $X_1,X_2, \ldots $ be independent uniform random points in a bounded region A ⊂ bf Rd having a smooth boundary, d ≥ 1. Let B ⊂ A be compact. The _coverage threshold_ of B, Rn, is the smallest r such that B is covered by the balls of radius r centred on X1,…,Xn. The _maximal spacing_ R̃n is the volume of the largest ball contained in A ∖ {X1,…,Xn}. We investigate the asymptotic frequency of _record times_ in the sequence (Rn), that is times n for which Rn < Rn−1. Let Nm denote the number of records in the sequence (Rn) up to time m, and let νm be the time at which the mth record value of the sequence (Rn) occurs. For B ⊂ Ao, we show that almost surely, Nn ∼ 1/2 (log n)2 and νn1/√n→ exp (√2 ) as n → ∞, and likewise for Ñn and ν̃n, defined analogously in terms of (R̃n). But if B=A and d ≥ 3, then Nn ∼ 1/2 (1− 1/d) (log n)2 and νn1/√n→ exp ( √(2d/(d−1)) ). We also discuss the generalization (for fixed k ∈ bf N) to k-coverage thresholds, maximal k-spacings and non-uniformly distributed points Xi in A.

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Cite

BibTeXRIS
BibTeX
@misc{penrose2026record,
  title = {{Record times for coverage thresholds and maximal spacings}},
  author = {Mathew D. Penrose},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.19104},
  eprint = {2608.19104},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.19104},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Record times for coverage thresholds and maximal spacings
AU  - Mathew D. Penrose
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2608.19104
UR  - https://arxiv.org/abs/2608.19104
AB  - Let $X_1,X_2, \ldots $ be independent uniform random points in a bounded region A ⊂ bf R^d having a smooth boundary, d ≥ 1. Let B ⊂ A be compact. The _coverage threshold_ of B, R_n, is the smallest r such that B is covered by the balls of radius r centred on X_1,…,X_n. The _maximal spacing_ R̃_n is the volume of the largest ball contained in A ∖ {X_1,…,X_n}. We investigate the asymptotic frequency of _record times_ in the sequence (R_n), that is times n for which R_n < R_n−1. Let N_m denote the number of records in the sequence (R_n) up to time m, and let ν_m be the time at which the mth record value of the sequence (R_n) occurs. For B ⊂ A^o, we show that almost surely, N_n ∼ 1/2 (log n)^2 and ν_n^1/√n→ exp (√2 ) as n → ∞, and likewise for Ñ_n and ν̃_n, defined analogously in terms of (R̃_n). But if B=A and d ≥ 3, then N_n ∼ 1/2 (1− 1/d) (log n)^2 and ν_n^1/√n→ exp ( √(2d/(d−1)) ). We also discuss the generalization (for fixed k ∈ bf N) to k-coverage thresholds, maximal k-spacings and non-uniformly distributed points X_i in A.
ER  -