Three Favorite Edges Occurs Infinitely Often for One-Dimensional Simple Random Walk

IF 1.1 4区数学 Q1 MATHEMATICS

Communications in Mathematics and Statistics Pub Date : 2024-07-17 DOI:10.1007/s40304-023-00382-2

Chen-Xu Hao, Ze-Chun Hu, Ting Ma, Renming Song

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).

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一维简单随机游走无限频繁地出现三个喜欢的边缘

对于一维简单对称随机游走（(S_n)\），如果一条边 x（在点 \(x-1\) 和 x 之间）在 n 时刻的局部时间在所有边中达到最大，那么这条边 x 在 n 时刻被称为最爱边。在本文中，我们证明了在概率为 1 的情况下，三条最喜欢的边会无限频繁地出现。我们的工作受到 Tóth 和 Werner（Comb Probab Comput 6:359-369, 1997）以及 Ding 和 Shen（Ann Probab 46:2545-2561, 2018）的启发，推翻了 Tóth 和 Werner（1997）第 368 页备注 1 中提到的一个猜想。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Communications in Mathematics and Statistics Mathematics-Statistics and Probability

CiteScore

1.80

自引率

0.00%

发文量

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.