{"title":"Three Favorite Edges Occurs Infinitely Often for One-Dimensional Simple Random Walk","authors":"Chen-Xu Hao, Ze-Chun Hu, Ting Ma, Renming Song","doi":"10.1007/s40304-023-00382-2","DOIUrl":null,"url":null,"abstract":"<p>For a one-dimensional simple symmetric random walk <span>\\((S_n)\\)</span>, an edge <i>x</i> (between points <span>\\(x-1\\)</span> and <i>x</i>) is called a favorite edge at time <i>n</i> if its local time at <i>n</i> achieves the maximum among all edges. In this paper, we show that with probability 1 three favorite edges occurs infinitely often. Our work is inspired by Tóth and Werner (Comb Probab Comput 6:359–369, 1997), and Ding and Shen (Ann Probab 46:2545–2561, 2018), disproves a conjecture mentioned in Remark 1 on page 368 of Tóth and Werner (1997).</p>","PeriodicalId":10575,"journal":{"name":"Communications in Mathematics and Statistics","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematics and Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40304-023-00382-2","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
For a one-dimensional simple symmetric random walk \((S_n)\), an edge x (between points \(x-1\) and x) is called a favorite edge at time n if its local time at n achieves the maximum among all edges. In this paper, we show that with probability 1 three favorite edges occurs infinitely often. Our work is inspired by Tóth and Werner (Comb Probab Comput 6:359–369, 1997), and Ding and Shen (Ann Probab 46:2545–2561, 2018), disproves a conjecture mentioned in Remark 1 on page 368 of Tóth and Werner (1997).
期刊介绍：
Communications in Mathematics and Statistics is an international journal published by Springer-Verlag in collaboration with the School of Mathematical Sciences, University of Science and Technology of China (USTC). The journal will be committed to publish high level original peer reviewed research papers in various areas of mathematical sciences, including pure mathematics, applied mathematics, computational mathematics, and probability and statistics. Typically one volume is published each year, and each volume consists of four issues.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
