Left-continuous random walk on $\mathbb{Z}$ and the parity of its hitting times
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916317700816896 |
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| author | Vilkas, Timo |
| author_facet | Vilkas, Timo |
| contents | When it comes to random walk on the integers $\mathbb{Z}$, the arguably first step of generalization beyond simple random walk is the class of one-sidedly continuous random walk, where the stepsize in only one direction is bounded by 1. Moreover, the time until state 0 is hit by left-continuous random walk on $\mathbb{Z}$ has a direct connection to the total progeny in branching processes. In this article, the probability of left-continuous random walk to be negative at an even (resp.\ odd) time is derived and used to determine the probability of nearly left-continuous random walk to eventually become negative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06903 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Left-continuous random walk on $\mathbb{Z}$ and the parity of its hitting times Vilkas, Timo Probability 60G50, 60J10 When it comes to random walk on the integers $\mathbb{Z}$, the arguably first step of generalization beyond simple random walk is the class of one-sidedly continuous random walk, where the stepsize in only one direction is bounded by 1. Moreover, the time until state 0 is hit by left-continuous random walk on $\mathbb{Z}$ has a direct connection to the total progeny in branching processes. In this article, the probability of left-continuous random walk to be negative at an even (resp.\ odd) time is derived and used to determine the probability of nearly left-continuous random walk to eventually become negative. |
| title | Left-continuous random walk on $\mathbb{Z}$ and the parity of its hitting times |
| topic | Probability 60G50, 60J10 |
| url | https://arxiv.org/abs/2407.06903 |