| 一维简单随机游动的最爱下穿点 |
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| 引用本文： | 郝晨旭,胡泽春,马婷,宋仁明. 一维简单随机游动的最爱下穿点[J]. 四川大学学报(自然科学版), 2023, 60(5): 051002-58 |
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| 作者姓名： | 郝晨旭 胡泽春 马婷 宋仁明 |
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| 作者单位： | 四川大学数学学院,四川大学数学学院,四川大学数学学院,美国伊利诺伊大学香槟分校数学系 |
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| 基金项目： | 国家自然科学基金（12171335, 12101429, 12071011, 11931004, 11871184）;西蒙斯基金（429343）;四川大学理科发展专项项目（2020SCUNL201） |
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| 摘 要： | 随机游动是一类重要马氏过程,有广泛应用.对于一维简单对称随机游动(Sn),点x在时间n被称为最爱下穿点,如果在时刻n没有其它点的下穿次数比点x的下穿次数大.本文研究了最爱下穿点集的个数,证明如下的命题以概率1成立：4个及4个以上的最爱下穿点只能出现有限次,3个最爱下穿点则会出现无穷多次.此外,本文还提出了一些相关的开问题.
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| 关 键 词： | 随机游动 最爱点 最爱边 最爱下穿点 局部时 |
| 收稿时间： | 2022-10-23 |
| 修稿时间： | 2022-12-01 |
Favorite down-crossing sites of one-dimensional simple random walk |
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| HAO Chen-Xu,HU Ze-Chun,MA Ting and SONG Ren-Ming. Favorite down-crossing sites of one-dimensional simple random walk[J]. Journal of Sichuan University (Natural Science Edition), 2023, 60(5): 051002-58 |
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| Authors: | HAO Chen-Xu HU Ze-Chun MA Ting SONG Ren-Ming |
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| Affiliation: | School of Mathematics, Sichuan University,School of Mathematics, Sichuan University,School of Mathematics, Sichuan University,Department of Mathematics, University of Illinois |
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| Abstract: | Random walk is a very important Markov process and has important applications in many fields. For a one-dimensional simple symmetric random walk, a site is called a favorite down-crossing site at a time if its down-crossing local time at the time achieves the maximum among all sites. In this paper, we study the cardinality of the favorite down-crossing site set, and show that with probability 1 there are only finitely many times at which there are at least four favorite down-crossing sites and three favorite down-crossing sites occurs infinitely often. Some related open questions are introduced as well. |
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