The asymptotic behavior of rarely visited edges of the simple random walk
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912832312836096 |
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| author | Hu, Ze-Chun Peng, Xue Song, Renming Tan, Yuan |
| author_facet | Hu, Ze-Chun Peng, Xue Song, Renming Tan, Yuan |
| contents | In this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that visited only once) of a simple symmetric random walk on $\mathbb{Z}$. Let $α(n)$ be the number of rarely visited edges up to time $n$. First, we evaluate $\mathbb{E}(α(n))$, show that $n\to \mathbb{E}(α(n))$ is non-decreasing in $n$ and that $\lim\limits_{n\to+\infty}\mathbb{E}(α(n))=2$. Then we study the asymptotic behavior of $\mathbb{P} (α(n)>a(\log n)^2)$ for any $a>0$ and use it to show that there exists a constant $C\in(1/32,1/2]$ such that $\limsup\limits_{n\to+\infty}\frac{α(n)}{(\log n)^2}=C$ almost surely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_16657 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The asymptotic behavior of rarely visited edges of the simple random walk Hu, Ze-Chun Peng, Xue Song, Renming Tan, Yuan Probability In this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that visited only once) of a simple symmetric random walk on $\mathbb{Z}$. Let $α(n)$ be the number of rarely visited edges up to time $n$. First, we evaluate $\mathbb{E}(α(n))$, show that $n\to \mathbb{E}(α(n))$ is non-decreasing in $n$ and that $\lim\limits_{n\to+\infty}\mathbb{E}(α(n))=2$. Then we study the asymptotic behavior of $\mathbb{P} (α(n)>a(\log n)^2)$ for any $a>0$ and use it to show that there exists a constant $C\in(1/32,1/2]$ such that $\limsup\limits_{n\to+\infty}\frac{α(n)}{(\log n)^2}=C$ almost surely. |
| title | The asymptotic behavior of rarely visited edges of the simple random walk |
| topic | Probability |
| url | https://arxiv.org/abs/2310.16657 |