{"title":"关于一些具有 Lipschitz 数字 1 的随机函数的迭代","authors":"Yingdong Lu, Tomasz Nowicki","doi":"arxiv-2409.06003","DOIUrl":null,"url":null,"abstract":"For the iterations of $x\\mapsto |x-\\theta|$ random functions with Lipschitz\nnumber one, we represent the dynamics as a Markov chain and prove its\nconvergence under mild conditions. We also demonstrate that the Wasserstein\nmetric of any two measures will not increase after the corresponding induced\niterations for measures and identify conditions under which a polynomial\nconvergence rate can be achieved in this metric. We also consider an associated\nnonlinear operator on the space of probability measures and identify its fixed\npoints through an detailed analysis of their characteristic functions.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"4 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the iterations of some random functions with Lipschitz number one\",\"authors\":\"Yingdong Lu, Tomasz Nowicki\",\"doi\":\"arxiv-2409.06003\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For the iterations of $x\\\\mapsto |x-\\\\theta|$ random functions with Lipschitz\\nnumber one, we represent the dynamics as a Markov chain and prove its\\nconvergence under mild conditions. We also demonstrate that the Wasserstein\\nmetric of any two measures will not increase after the corresponding induced\\niterations for measures and identify conditions under which a polynomial\\nconvergence rate can be achieved in this metric. We also consider an associated\\nnonlinear operator on the space of probability measures and identify its fixed\\npoints through an detailed analysis of their characteristic functions.\",\"PeriodicalId\":501245,\"journal\":{\"name\":\"arXiv - MATH - Probability\",\"volume\":\"4 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Probability\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.06003\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06003","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the iterations of some random functions with Lipschitz number one
For the iterations of $x\mapsto |x-\theta|$ random functions with Lipschitz
number one, we represent the dynamics as a Markov chain and prove its
convergence under mild conditions. We also demonstrate that the Wasserstein
metric of any two measures will not increase after the corresponding induced
iterations for measures and identify conditions under which a polynomial
convergence rate can be achieved in this metric. We also consider an associated
nonlinear operator on the space of probability measures and identify its fixed
points through an detailed analysis of their characteristic functions.