The Exact Limsup Constant for Once-Visited Sites of One-Dimensional Simple Random Walk
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917384551399424 |
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| author | Feng, Chenxu Hao, Chenxu |
| author_facet | Feng, Chenxu Hao, Chenxu |
| contents | For a one-dimensional simple random walk, let $g_1(n)$ denote the number of sites visited exactly once at time $n$. Major (1988) proved that \begin{equation*} \limsup_{n\to\infty}\frac{g_1(n)}{\log^2 n}=C\qquad a.s. \end{equation*} where $C$ is a positive and finite constant. While this result settled the question of existence, the exact value of $C$ remained unknown.
In this paper, we determine that $C=1/16$. The main novelty of our work lies in introducing a self-boosting iterative framework for analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21602 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Exact Limsup Constant for Once-Visited Sites of One-Dimensional Simple Random Walk Feng, Chenxu Hao, Chenxu Probability 60F15, 60J55 For a one-dimensional simple random walk, let $g_1(n)$ denote the number of sites visited exactly once at time $n$. Major (1988) proved that \begin{equation*} \limsup_{n\to\infty}\frac{g_1(n)}{\log^2 n}=C\qquad a.s. \end{equation*} where $C$ is a positive and finite constant. While this result settled the question of existence, the exact value of $C$ remained unknown. In this paper, we determine that $C=1/16$. The main novelty of our work lies in introducing a self-boosting iterative framework for analysis. |
| title | The Exact Limsup Constant for Once-Visited Sites of One-Dimensional Simple Random Walk |
| topic | Probability 60F15, 60J55 |
| url | https://arxiv.org/abs/2511.21602 |