{"title":"Recurrence and transience of Rademacher series","authors":"Satyaki Bhattacharya, S. Volkov","doi":"10.30757/alea.v20-03","DOIUrl":null,"url":null,"abstract":"We introduce the notion of {\\bf a}-walk $S(n)=a_1 X_1+\\dots+a_n X_n$, based on a sequence of positive numbers ${\\bf a}=(a_1,a_2,\\dots)$ and a Rademacher sequence $X_1,X_2,\\dots$. We study recurrence/transience (properly defined) of such walks for various sequences of ${\\bf a}$. In particular, we establish the classification in the cases where $a_k=\\lfloor k^\\beta\\rfloor$, $\\beta>0$, as well as in the case $a_k=\\lceil \\log_\\gamma k \\rceil$ or $a_k=\\log_\\gamma k$ for $\\gamma>1$.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2022-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Alea-Latin American Journal of Probability and Mathematical Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.30757/alea.v20-03","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
We introduce the notion of {\bf a}-walk $S(n)=a_1 X_1+\dots+a_n X_n$, based on a sequence of positive numbers ${\bf a}=(a_1,a_2,\dots)$ and a Rademacher sequence $X_1,X_2,\dots$. We study recurrence/transience (properly defined) of such walks for various sequences of ${\bf a}$. In particular, we establish the classification in the cases where $a_k=\lfloor k^\beta\rfloor$, $\beta>0$, as well as in the case $a_k=\lceil \log_\gamma k \rceil$ or $a_k=\log_\gamma k$ for $\gamma>1$.
期刊介绍:
ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted.
ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper.
ALEA is affiliated with the Institute of Mathematical Statistics.