{"title":"随机环境中分支过程的中心极限定理精确收敛率和多项式收敛率","authors":"Yingqiu Li , Xin Zhang , Zhan Lu , Sheng Xiao","doi":"10.1016/j.spl.2024.110268","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mrow><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></math></span> be a supercritical branching process in an independent and identically distributed (i.i.d.) random environment. The paper studies the properties of the estimator <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mrow><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> introduced by Dion and Esty in 1979. We introduce a related martingale and discuss its convergence and exponential convergence rate. On this basis the exact convergence rate of the central limit theorem for normalized <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is given.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167715224002372/pdfft?md5=220bf7e97e493e929a1b6a021826f150&pid=1-s2.0-S0167715224002372-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Exact convergence rate of the central limit theorem and polynomial convergence rate for branching processes in a random environment\",\"authors\":\"Yingqiu Li , Xin Zhang , Zhan Lu , Sheng Xiao\",\"doi\":\"10.1016/j.spl.2024.110268\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <span><math><mrow><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></math></span> be a supercritical branching process in an independent and identically distributed (i.i.d.) random environment. The paper studies the properties of the estimator <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mrow><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> introduced by Dion and Esty in 1979. We introduce a related martingale and discuss its convergence and exponential convergence rate. On this basis the exact convergence rate of the central limit theorem for normalized <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is given.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-09-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0167715224002372/pdfft?md5=220bf7e97e493e929a1b6a021826f150&pid=1-s2.0-S0167715224002372-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0167715224002372\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167715224002372","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Exact convergence rate of the central limit theorem and polynomial convergence rate for branching processes in a random environment
Let be a supercritical branching process in an independent and identically distributed (i.i.d.) random environment. The paper studies the properties of the estimator introduced by Dion and Esty in 1979. We introduce a related martingale and discuss its convergence and exponential convergence rate. On this basis the exact convergence rate of the central limit theorem for normalized is given.
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