{"title":"用一个点过程覆盖另一个点过程。","authors":"Frankie Higgs, Mathew D Penrose, Xiaochuan Yang","doi":"10.1007/s11009-025-10165-7","DOIUrl":null,"url":null,"abstract":"<p><p>Let <math> <mrow><msub><mi>X</mi> <mn>1</mn></msub> <mo>,</mo> <msub><mi>X</mi> <mn>2</mn></msub> <mo>,</mo> <mo>…</mo></mrow> </math> and <math> <mrow><msub><mi>Y</mi> <mn>1</mn></msub> <mo>,</mo> <msub><mi>Y</mi> <mn>2</mn></msub> <mo>,</mo> <mo>…</mo></mrow> </math> be i.i.d. random uniform points in a bounded domain <math><mrow><mi>A</mi> <mo>⊂</mo> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> </mrow> </math> with smooth or polygonal boundary. Given <math><mrow><mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>k</mi> <mo>∈</mo> <mi>N</mi></mrow> </math> , define the <i>two-sample k-coverage threshold</i> <math><msub><mi>R</mi> <mrow><mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>k</mi></mrow> </msub> </math> to be the smallest <i>r</i> such that each point of <math><mrow><mo>{</mo> <msub><mi>Y</mi> <mn>1</mn></msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub><mi>Y</mi> <mi>m</mi></msub> <mo>}</mo></mrow> </math> is covered at least <i>k</i> times by the disks of radius <i>r</i> centred on <math> <mrow><msub><mi>X</mi> <mn>1</mn></msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub><mi>X</mi> <mi>n</mi></msub> </mrow> </math> . We obtain the limiting distribution of <math><msub><mi>R</mi> <mrow><mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>k</mi></mrow> </msub> </math> as <math><mrow><mi>n</mi> <mo>→</mo> <mi>∞</mi></mrow> </math> with <math><mrow><mi>m</mi> <mo>=</mo> <mi>m</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> <mo>∼</mo> <mi>τ</mi> <mi>n</mi></mrow> </math> for some constant <math><mrow><mi>τ</mi> <mo>></mo> <mn>0</mn></mrow> </math> , with <i>k</i> fixed. If <i>A</i> has unit area, then <math><mrow><mi>n</mi> <mi>π</mi> <msubsup><mi>R</mi> <mrow><mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> <mo>,</mo> <mn>1</mn></mrow> <mn>2</mn></msubsup> <mo>-</mo> <mo>log</mo> <mi>n</mi></mrow> </math> is asymptotically Gumbel distributed with scale parameter 1 and location parameter <math><mrow><mo>log</mo> <mi>τ</mi></mrow> </math> . For <math><mrow><mi>k</mi> <mo>></mo> <mn>2</mn></mrow> </math> , we find that <math><mrow><mi>n</mi> <mi>π</mi> <msubsup><mi>R</mi> <mrow><mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> <mo>,</mo> <mi>k</mi></mrow> <mn>2</mn></msubsup> <mo>-</mo> <mo>log</mo> <mi>n</mi> <mo>-</mo> <mrow><mo>(</mo> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>3</mn> <mo>)</mo></mrow> <mo>log</mo> <mo>log</mo> <mi>n</mi></mrow> </math> is asymptotically Gumbel with scale parameter 2 and a more complicated location parameter involving the perimeter of <i>A</i>; boundary effects dominate when <math><mrow><mi>k</mi> <mo>></mo> <mn>2</mn></mrow> </math> . For <math><mrow><mi>k</mi> <mo>=</mo> <mn>2</mn></mrow> </math> 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 <i>k</i>.</p>","PeriodicalId":18442,"journal":{"name":"Methodology and Computing in Applied Probability","volume":"27 2","pages":"40"},"PeriodicalIF":1.0000,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12041118/pdf/","citationCount":"0","resultStr":"{\"title\":\"Covering One Point Process with Another.\",\"authors\":\"Frankie Higgs, Mathew D Penrose, Xiaochuan Yang\",\"doi\":\"10.1007/s11009-025-10165-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>Let <math> <mrow><msub><mi>X</mi> <mn>1</mn></msub> <mo>,</mo> <msub><mi>X</mi> <mn>2</mn></msub> <mo>,</mo> <mo>…</mo></mrow> </math> and <math> <mrow><msub><mi>Y</mi> <mn>1</mn></msub> <mo>,</mo> <msub><mi>Y</mi> <mn>2</mn></msub> <mo>,</mo> <mo>…</mo></mrow> </math> be i.i.d. random uniform points in a bounded domain <math><mrow><mi>A</mi> <mo>⊂</mo> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> </mrow> </math> with smooth or polygonal boundary. Given <math><mrow><mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>k</mi> <mo>∈</mo> <mi>N</mi></mrow> </math> , define the <i>two-sample k-coverage threshold</i> <math><msub><mi>R</mi> <mrow><mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>k</mi></mrow> </msub> </math> to be the smallest <i>r</i> such that each point of <math><mrow><mo>{</mo> <msub><mi>Y</mi> <mn>1</mn></msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub><mi>Y</mi> <mi>m</mi></msub> <mo>}</mo></mrow> </math> is covered at least <i>k</i> times by the disks of radius <i>r</i> centred on <math> <mrow><msub><mi>X</mi> <mn>1</mn></msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub><mi>X</mi> <mi>n</mi></msub> </mrow> </math> . We obtain the limiting distribution of <math><msub><mi>R</mi> <mrow><mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>k</mi></mrow> </msub> </math> as <math><mrow><mi>n</mi> <mo>→</mo> <mi>∞</mi></mrow> </math> with <math><mrow><mi>m</mi> <mo>=</mo> <mi>m</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> <mo>∼</mo> <mi>τ</mi> <mi>n</mi></mrow> </math> for some constant <math><mrow><mi>τ</mi> <mo>></mo> <mn>0</mn></mrow> </math> , with <i>k</i> fixed. If <i>A</i> has unit area, then <math><mrow><mi>n</mi> <mi>π</mi> <msubsup><mi>R</mi> <mrow><mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> <mo>,</mo> <mn>1</mn></mrow> <mn>2</mn></msubsup> <mo>-</mo> <mo>log</mo> <mi>n</mi></mrow> </math> is asymptotically Gumbel distributed with scale parameter 1 and location parameter <math><mrow><mo>log</mo> <mi>τ</mi></mrow> </math> . For <math><mrow><mi>k</mi> <mo>></mo> <mn>2</mn></mrow> </math> , we find that <math><mrow><mi>n</mi> <mi>π</mi> <msubsup><mi>R</mi> <mrow><mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> <mo>,</mo> <mi>k</mi></mrow> <mn>2</mn></msubsup> <mo>-</mo> <mo>log</mo> <mi>n</mi> <mo>-</mo> <mrow><mo>(</mo> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>3</mn> <mo>)</mo></mrow> <mo>log</mo> <mo>log</mo> <mi>n</mi></mrow> </math> is asymptotically Gumbel with scale parameter 2 and a more complicated location parameter involving the perimeter of <i>A</i>; boundary effects dominate when <math><mrow><mi>k</mi> <mo>></mo> <mn>2</mn></mrow> </math> . For <math><mrow><mi>k</mi> <mo>=</mo> <mn>2</mn></mrow> </math> 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 <i>k</i>.</p>\",\"PeriodicalId\":18442,\"journal\":{\"name\":\"Methodology and Computing in Applied Probability\",\"volume\":\"27 2\",\"pages\":\"40\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2025-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12041118/pdf/\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Methodology and Computing in Applied Probability\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s11009-025-10165-7\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2025/4/29 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q3\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Methodology and Computing in Applied Probability","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11009-025-10165-7","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/4/29 0:00:00","PubModel":"Epub","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
摘要
设X 1, X 2,…和Y 1, Y 2,…是边界光滑或多边形的有界域a∧R 2中的随机一致点。给定n, m, k∈n,定义两样本k覆盖阈值R n, m, k为最小R,使得{Y 1,…,Y m}的每个点被以X 1,…,X n为中心的半径为R的磁盘覆盖至少k次。我们得到了R n, m, k为n→∞且m = m (n) ~ τ n时,对于某常数τ > 0, k固定的极限分布。如果A具有单位面积,则n π R n, m (n), 1,2 - log n是尺度参数为1,位置参数为log τ的渐近Gumbel分布。对于k > 2,我们发现n π R n, m (n), k 2 - log n - (2k - 3) log log n是具有尺度参数2和更复杂的位置参数涉及a周长的渐近Gumbel;当k > 2时,边界效应起主导作用。当k = 2时,极限cdf是一个尺度参数为1和2的双分量极值分布。我们也给出了高维的类似结果,其中边界效应在所有k中占主导地位。
Let and be i.i.d. random uniform points in a bounded domain with smooth or polygonal boundary. Given , define the two-sample k-coverage threshold to be the smallest r such that each point of is covered at least k times by the disks of radius r centred on . We obtain the limiting distribution of as with for some constant , with k fixed. If A has unit area, then is asymptotically Gumbel distributed with scale parameter 1 and location parameter . For , we find that is asymptotically Gumbel with scale parameter 2 and a more complicated location parameter involving the perimeter of A; boundary effects dominate when . For 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.
期刊介绍:
Methodology and Computing in Applied Probability will publish high quality research and review articles in the areas of applied probability that emphasize methodology and computing. Of special interest are articles in important areas of applications that include detailed case studies. Applied probability is a broad research area that is of interest to many scientists in diverse disciplines including: anthropology, biology, communication theory, economics, epidemiology, finance, linguistics, meteorology, operations research, psychology, quality control, reliability theory, sociology and statistics.
The following alphabetical listing of topics of interest to the journal is not intended to be exclusive but to demonstrate the editorial policy of attracting papers which represent a broad range of interests:
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Approximations-
Asymptotic Approximations & Expansions-
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Communication Networks-
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Information Theory-
Mathematical Physics-
Molecular Biology-
Monte Carlo Methods-
Order Statistics-
Queuing Theory-
Reliability Theory-
Stochastic Processes