{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/covering-one-point-process-with-another","title":"Covering one point process with another","arxiv_id":"2401.03832","date":"2024-01-08","proceeding":null,"authors":["Frankie Higgs","Mathew D. Penrose","Xiaochuan Yang"],"abstract":"Let $X_1,X_2, \\ldots $ and $Y_1, Y_2, \\ldots$ be i.i.d. random uniform points in a bounded domain $A \\subset \\mathbb{R}^2$ with smooth or polygonal boundary. Given $n,m,k \\in \\mathbb{N}$, define the {\\em two-sample $k$-coverage threshold} $R_{n,m,k}$ to be the smallest $r$ such that each point of $ \\{Y_1,\\ldots,Y_m\\}$ is covered at least $k$ times by the disks of radius $r$ centred on $X_1,\\ldots,X_n$. We obtain the limiting distribution of $R_{n,m,k}$ as $n \\to \\infty$ with $m= m(n) \\sim \\tau n$ for some constant $\\tau >0$, with $k $ fixed. If $A$ has unit area, then $n \\pi R_{n,m(n),1}^2 - \\log n$ is asymptotically Gumbel distributed with scale parameter $1$ and location parameter $\\log \\tau$. For $k >2$, we find that $n \\pi R_{n,m(n),k}^2 - \\log n - (2k-3) \\log \\log n$ is asymptotically Gumbel with scale parameter $2$ and a more complicated location parameter involving the perimeter of $A$; boundary effects dominate when $k >2$. For $k=2$ the limiting cdf is a two-component extreme value distribution with scale parameters 1 and 2. We also give analogous results for higher dimensions, where the boundary effects dominate for all $k$.","url_abs":"https://arxiv.org/abs/2401.03832v2","url_pdf":"https://arxiv.org/pdf/2401.03832v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"covering-one-point-process-with-another","repo_url":"https://github.com/frankiehiggs/covxy","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}