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Covering One Point Process with Another.

Frankie Higgs1, Mathew D Penrose1, Xiaochuan Yang2

  • 1Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY UK.

Methodology and Computing in Applied Probability
|May 2, 2025
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Summary
This summary is machine-generated.

This study analyzes the two-sample k-coverage threshold for random points in a domain. Results show its limiting distribution depends on domain area and perimeter, with boundary effects significant in higher dimensions.

Keywords:
Coverage thresholdPoisson point processWeak limit

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Area of Science:

  • Geometric probability
  • Spatial statistics
  • Extreme value theory

Background:

  • Understanding random point coverage is crucial in various fields.
  • The behavior of coverage thresholds in bounded domains requires detailed analysis.

Purpose of the Study:

  • To determine the limiting distribution of the two-sample k-coverage threshold.
  • To investigate the influence of domain geometry (area and boundary) on coverage.
  • To extend findings to higher dimensions.

Main Methods:

  • Utilizing probability theory and stochastic geometry.
  • Deriving asymptotic distributions for the coverage threshold.
  • Analyzing boundary effects in two and higher dimensions.

Main Results:

  • For unit area domains, the threshold follows a Gumbel distribution.
  • For domains with non-unit area, perimeter effects become significant, altering the distribution.
  • Higher dimensions show boundary effects dominating for all k.

Conclusions:

  • The geometric properties of the domain critically influence coverage thresholds.
  • Boundary effects are more pronounced in higher-dimensional spaces.
  • The derived distributions provide a theoretical framework for spatial coverage analysis.