Elephant random walk with polynomially decaying steps
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915268267081728 |
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| author | Nakano, Yuzaburo |
| author_facet | Nakano, Yuzaburo |
| contents | In this paper, we introduce a variation of the elephant random walk whose steps are polynomially decaying. At each time $k$, the walker's step size is $k^{-γ}$ with $γ>0$. We investigate effects of the step size exponent $γ$ and the memory parameter $α\in [-1,1]$ on the long-time behavior of the walker. For fixed $α$, it admits phase transition from divergence to convergence (localization) at $γ_{c}(α)=\max \{α,1/2\}$. This means that large enough memory effect can shift the critical point for localization. Moreover, we obtain quantitative limit theorems which provide a detailed picture of the long-time behavior of the walker. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00277 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Elephant random walk with polynomially decaying steps Nakano, Yuzaburo Probability In this paper, we introduce a variation of the elephant random walk whose steps are polynomially decaying. At each time $k$, the walker's step size is $k^{-γ}$ with $γ>0$. We investigate effects of the step size exponent $γ$ and the memory parameter $α\in [-1,1]$ on the long-time behavior of the walker. For fixed $α$, it admits phase transition from divergence to convergence (localization) at $γ_{c}(α)=\max \{α,1/2\}$. This means that large enough memory effect can shift the critical point for localization. Moreover, we obtain quantitative limit theorems which provide a detailed picture of the long-time behavior of the walker. |
| title | Elephant random walk with polynomially decaying steps |
| topic | Probability |
| url | https://arxiv.org/abs/2505.00277 |