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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.03873 |
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| _version_ | 1866915670453649408 |
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| author | Jin, Bochen |
| author_facet | Jin, Bochen |
| contents | We prove the limit theorem for paths of random walks with $n$ steps in $\mathbb{R}^d$ as $n$ and $d$ both go to infinity. For this, the paths are viewed as finite metric spaces equipped with the $\ell_p$-metric for $p\in[1,\infty)$. Under the assumptions that all components of each step are uncorrelated, centered, have finite $2p$-th moments, and are identically distributed, we show that such random metric space converges in probability to a deterministic limit space with respect to the Gromov-Hausdorff distance. This result generalises earlier work by Kabluchko and Marynych for $p=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03873 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence of Random Walks in $\ell_p$-Spaces of Growing Dimension Jin, Bochen Probability 60F05, 60G50 We prove the limit theorem for paths of random walks with $n$ steps in $\mathbb{R}^d$ as $n$ and $d$ both go to infinity. For this, the paths are viewed as finite metric spaces equipped with the $\ell_p$-metric for $p\in[1,\infty)$. Under the assumptions that all components of each step are uncorrelated, centered, have finite $2p$-th moments, and are identically distributed, we show that such random metric space converges in probability to a deterministic limit space with respect to the Gromov-Hausdorff distance. This result generalises earlier work by Kabluchko and Marynych for $p=2$. |
| title | Convergence of Random Walks in $\ell_p$-Spaces of Growing Dimension |
| topic | Probability 60F05, 60G50 |
| url | https://arxiv.org/abs/2512.03873 |