Covering one point process with another
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909456669868032 |
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| author | Higgs, Frankie Penrose, Mathew D. Yang, Xiaochuan |
| author_facet | Higgs, Frankie Penrose, Mathew D. Yang, Xiaochuan |
| contents | 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 $m= m(n) \sim τn$ for some constant $τ>0$, with $k $ fixed. If $A$ has unit area, then $n πR_{n,m(n),1}^2 - \log n$ is asymptotically Gumbel distributed with scale parameter $1$ and location parameter $\log τ$. For $k >2$, we find that $n πR_{n,m(n),k}^2 - \log n - (2k-3) \log \log n$ 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 $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03832 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Covering one point process with another Higgs, Frankie Penrose, Mathew D. Yang, Xiaochuan Probability 60D05, 60F05, 60F15 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 $m= m(n) \sim τn$ for some constant $τ>0$, with $k $ fixed. If $A$ has unit area, then $n πR_{n,m(n),1}^2 - \log n$ is asymptotically Gumbel distributed with scale parameter $1$ and location parameter $\log τ$. For $k >2$, we find that $n πR_{n,m(n),k}^2 - \log n - (2k-3) \log \log n$ 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 $k$. |
| title | Covering one point process with another |
| topic | Probability 60D05, 60F05, 60F15 |
| url | https://arxiv.org/abs/2401.03832 |