{"title":"Recurrence and transience of random difference equations in the critical case","authors":"G. Alsmeyer, A. Iksanov","doi":"10.1214/22-aihp1274","DOIUrl":null,"url":null,"abstract":"For i.i.d. random vectors $(M_{1},Q_{1}),(M_{2},Q_{2}),\\ldots$ such that $M>0$ a.s., $Q\\geq 0$ a.s. and $\\mathbb{P}(Q=0)<1$, the random difference equation $X_{n}=M_{n}X_{n-1}+Q_{n}$, $n=1,2,\\ldots$, is studied in the critical case when the random walk with increments $\\log M_{1},\\log M_{2}$ is oscillating. We provide conditions for the null-recurrence and transience of the Markov chain $(X_{n})_{n\\ge 0}$ by inter alia drawing on techniques developed in the related article Alsmeyer et al (2017) for another case exhibiting the null-recurrence/transience dichotomy.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"46 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2021-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales de l Institut Henri Poincare D","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/22-aihp1274","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 1
Abstract
For i.i.d. random vectors $(M_{1},Q_{1}),(M_{2},Q_{2}),\ldots$ such that $M>0$ a.s., $Q\geq 0$ a.s. and $\mathbb{P}(Q=0)<1$, the random difference equation $X_{n}=M_{n}X_{n-1}+Q_{n}$, $n=1,2,\ldots$, is studied in the critical case when the random walk with increments $\log M_{1},\log M_{2}$ is oscillating. We provide conditions for the null-recurrence and transience of the Markov chain $(X_{n})_{n\ge 0}$ by inter alia drawing on techniques developed in the related article Alsmeyer et al (2017) for another case exhibiting the null-recurrence/transience dichotomy.