Recurrence and transience of random difference equations in the critical case

IF 1.5 Q2 PHYSICS, MATHEMATICAL
G. Alsmeyer, A. Iksanov
{"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.
临界情况下随机差分方程的递归性和暂态性
对于i.i.d随机向量$(M_{1},Q_{1}),(M_{2},Q_{2}),\ldots$,如$M>0$ a.s., $Q\geq 0$ a.s.和$\mathbb{P}(Q=0)<1$,研究了随机差分方程$X_{n}=M_{n}X_{n-1}+Q_{n}$, $n=1,2,\ldots$,在具有增量$\log M_{1},\log M_{2}$的随机游走振荡的临界情况下。我们为马尔可夫链$(X_{n})_{n\ge 0}$的零递归和瞬态提供了条件,除其他外,我们还利用了相关文章Alsmeyer等(2017)中开发的技术,用于另一种显示零递归/瞬态二分法的情况。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
CiteScore
2.30
自引率
0.00%
发文量
16
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信