Convergence of scaled asymptotically-free self-interacting random walks to Brownian motion perturbed at extrema
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| Format: | Preprint |
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2024
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| _version_ | 1866912299665588224 |
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| author | Liu, Xiaoyu Wang, Zhe |
| author_facet | Liu, Xiaoyu Wang, Zhe |
| contents | We consider a family of one-dimensional self interacting walks whose dynamics characterized by a monotone weight function $w$ on $\mathbb{N}\cup \{0\}$. The weight function takes the form $w(n) = (1 + 2^p Bn^{-p} + O(n^{-1-κ}))^{-1}$, for some $B \in \mathbb{R} $, $κ>0$ and $p\in (0,1]$. Our main model parameter is $p$, and for $p\in (0,1/2]$ we show the convergence of the SIRW to Brownian motion perturbed at extrema under the diffusive scaling. This completes the functional limit theorem in [8] for the asymptotically free case and extends the result to the full parameter range $(0,1]$. Our method depends on the generalized Ray-Knight theorems ([T96], [KMP23]) for the rescaled local times of this walk. The directed edge local times, described by the branching-like processes, are used to analyze the total drift experienced by the walker. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_11828 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of scaled asymptotically-free self-interacting random walks to Brownian motion perturbed at extrema Liu, Xiaoyu Wang, Zhe Probability 60K35 (Primary) 60F17, 60J85 (Secondary) We consider a family of one-dimensional self interacting walks whose dynamics characterized by a monotone weight function $w$ on $\mathbb{N}\cup \{0\}$. The weight function takes the form $w(n) = (1 + 2^p Bn^{-p} + O(n^{-1-κ}))^{-1}$, for some $B \in \mathbb{R} $, $κ>0$ and $p\in (0,1]$. Our main model parameter is $p$, and for $p\in (0,1/2]$ we show the convergence of the SIRW to Brownian motion perturbed at extrema under the diffusive scaling. This completes the functional limit theorem in [8] for the asymptotically free case and extends the result to the full parameter range $(0,1]$. Our method depends on the generalized Ray-Knight theorems ([T96], [KMP23]) for the rescaled local times of this walk. The directed edge local times, described by the branching-like processes, are used to analyze the total drift experienced by the walker. |
| title | Convergence of scaled asymptotically-free self-interacting random walks to Brownian motion perturbed at extrema |
| topic | Probability 60K35 (Primary) 60F17, 60J85 (Secondary) |
| url | https://arxiv.org/abs/2402.11828 |