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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
Summary
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.
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.
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