{"title":"二维类覆盖问题的大数定律","authors":"E. Ceyhan, J. Wierman, Pengfei Xiang","doi":"10.1051/ps/2021013","DOIUrl":null,"url":null,"abstract":"We prove a Law of Large Numbers (LLN) for the domination number of class cover catch digraphs (CCCD) generated by random points in two (or higher) dimensions. DeVinney and Wierman (2002) proved the Strong Law of Large Numbers (SLLN) for the uniform distribution in one dimension, and Wierman and Xiang (2008) extended the SLLN to the case of general distributions in one dimension. In this article, using subadditive processes, we prove a SLLN result for the domination number generated by Poisson points in ℝ2. From this we obtain a Weak Law of Large Numbers (WLLN) for the domination number generated by random points in [0, 1]2 from uniform distribution first, and then extend these result to the case of bounded continuous distributions. We also extend the results to higher dimensions. The domination number of CCCDs and related digraphs have applications in statistical pattern classification and spatial data analysis.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Law of large numbers for a two-dimensional class cover problem\",\"authors\":\"E. Ceyhan, J. Wierman, Pengfei Xiang\",\"doi\":\"10.1051/ps/2021013\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove a Law of Large Numbers (LLN) for the domination number of class cover catch digraphs (CCCD) generated by random points in two (or higher) dimensions. DeVinney and Wierman (2002) proved the Strong Law of Large Numbers (SLLN) for the uniform distribution in one dimension, and Wierman and Xiang (2008) extended the SLLN to the case of general distributions in one dimension. In this article, using subadditive processes, we prove a SLLN result for the domination number generated by Poisson points in ℝ2. From this we obtain a Weak Law of Large Numbers (WLLN) for the domination number generated by random points in [0, 1]2 from uniform distribution first, and then extend these result to the case of bounded continuous distributions. We also extend the results to higher dimensions. The domination number of CCCDs and related digraphs have applications in statistical pattern classification and spatial data analysis.\",\"PeriodicalId\":51249,\"journal\":{\"name\":\"Esaim-Probability and Statistics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2021-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Esaim-Probability and Statistics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1051/ps/2021013\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Esaim-Probability and Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1051/ps/2021013","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 1
摘要
我们证明了一类覆盖捕获有向图(CCCD)的支配数的大数定律(LLN)，这些有向图是由两个(或更高)维的随机点生成的。DeVinney和Wierman(2002)证明了一维均匀分布的强大数定律(Strong Law of Large Numbers, SLLN)， Wierman和Xiang(2008)将SLLN推广到一维一般分布的情况。本文利用次加性过程，证明了由泊松点生成的控制数的一个SLLN结果。由此首先得到了均匀分布下[0,1]2中随机点生成的支配数的一个弱大数定律，并将此结果推广到有界连续分布的情况。我们还将结果扩展到更高的维度。cccd及其相关有向图的支配数在统计模式分类和空间数据分析中具有广泛的应用。
Law of large numbers for a two-dimensional class cover problem
We prove a Law of Large Numbers (LLN) for the domination number of class cover catch digraphs (CCCD) generated by random points in two (or higher) dimensions. DeVinney and Wierman (2002) proved the Strong Law of Large Numbers (SLLN) for the uniform distribution in one dimension, and Wierman and Xiang (2008) extended the SLLN to the case of general distributions in one dimension. In this article, using subadditive processes, we prove a SLLN result for the domination number generated by Poisson points in ℝ2. From this we obtain a Weak Law of Large Numbers (WLLN) for the domination number generated by random points in [0, 1]2 from uniform distribution first, and then extend these result to the case of bounded continuous distributions. We also extend the results to higher dimensions. The domination number of CCCDs and related digraphs have applications in statistical pattern classification and spatial data analysis.
期刊介绍：
The journal publishes original research and survey papers in the area of Probability and Statistics. It covers theoretical and practical aspects, in any field of these domains.
Of particular interest are methodological developments with application in other scientific areas, for example Biology and Genetics, Information Theory, Finance, Bioinformatics, Random structures and Random graphs, Econometrics, Physics.
Long papers are very welcome.
Indeed, we intend to develop the journal in the direction of applications and to open it to various fields where random mathematical modelling is important. In particular we will call (survey) papers in these areas, in order to make the random community aware of important problems of both theoretical and practical interest. We all know that many recent fascinating developments in Probability and Statistics are coming from "the outside" and we think that ESAIM: P&S should be a good entry point for such exchanges. Of course this does not mean that the journal will be only devoted to practical aspects.
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