{"title":"Rates of convergence in the strong law of large numbers for weighted averages of nonidentically distributed random variables","authors":"","doi":"10.1007/s10986-024-09621-7","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>Integral tests are found for the convergence of two Spitzer-type series associated with a class of weighted averages introduced by Jajte [On the strong law of large numbers, <em>Ann. Probab.</em>, 31(1):409–412, 2003]. Our main theorems are valid for a large family of dependent random variables that are not necessarily identically distributed. As a byproduct, we improve the Marcinkiewicz–Zygmund strong law of large numbers for asymptotically almost negatively associated sequences due to Chandra and Ghosal [Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables <em>Acta Math. Hung.</em>, 71(4):327–336, 1996]. We also complement two limit theorems recently derived by Anh et al. [TheMarcinkiewicz–Zygmund-type strong law of large numbers with general normalizing sequences, <em>J. Theor. Probab.</em>, 34(1):331–348, 2021] and Thành [On a new concept of stochastic domination and the laws of large numbers, <em>Test</em>, 32(1):74–106, 2023]. The obtained results are new even when the summands are independent.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10986-024-09621-7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Integral tests are found for the convergence of two Spitzer-type series associated with a class of weighted averages introduced by Jajte [On the strong law of large numbers, Ann. Probab., 31(1):409–412, 2003]. Our main theorems are valid for a large family of dependent random variables that are not necessarily identically distributed. As a byproduct, we improve the Marcinkiewicz–Zygmund strong law of large numbers for asymptotically almost negatively associated sequences due to Chandra and Ghosal [Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables Acta Math. Hung., 71(4):327–336, 1996]. We also complement two limit theorems recently derived by Anh et al. [TheMarcinkiewicz–Zygmund-type strong law of large numbers with general normalizing sequences, J. Theor. Probab., 34(1):331–348, 2021] and Thành [On a new concept of stochastic domination and the laws of large numbers, Test, 32(1):74–106, 2023]. The obtained results are new even when the summands are independent.
摘要 对与 Jajte [《论强大数定律》，Ann. Probab.，31(1):409-412, 2003] 引入的一类加权平均数相关的两个 Spitzer 型数列的收敛性进行了积分检验。我们的主要定理适用于不一定是同分布的一大系列因变量。作为副产品，我们改进了 Chandra 和 Ghosal [Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables Acta Math.71(4):327-336, 1996].我们还补充了 Anh 等人最近推导的两个极限定理 [TheMarcinkiewicz-Zygmund-type strong law of large numbers with general normalizing sequences, J. Theor.Probab., 34(1):331-348, 2021] 和 Thành [On a new concept of stochastic domination and the laws of large numbers, Test, 32(1):74-106, 2023]。即使求和是独立的，所得到的结果也是新的。
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