Conditioned random walks on linear groups II: local limit theorems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908450141765632 |
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| author | Grama, Ion Quint, Jean-François Xiao, Hui |
| author_facet | Grama, Ion Quint, Jean-François Xiao, Hui |
| contents | We investigate random walks on the general linear group constrained within a specific domain, with a focus on their asymptotic behavior. In a previous work [38], we constructed the associated harmonic measure, a key element in formulating the local limit theorem for conditioned random walks on groups. The primary aim of this paper is to prove this theorem. The main challenge arises from studying the conditioned reverse walk, whose increments, in the context of random walks on groups, depend on the entire future. To achieve our goal, we combine a Caravenna-type conditioned local limit theorem with the conditioned version of the central limit theorem for the reversed walk. The resulting local limit theorem is then applied to derive the local behavior of the exit time. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_05897 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conditioned random walks on linear groups II: local limit theorems Grama, Ion Quint, Jean-François Xiao, Hui Probability Dynamical Systems Primary 60B15, 60B20, 60F05, 60J05. Secondary 22D40, 22E46 We investigate random walks on the general linear group constrained within a specific domain, with a focus on their asymptotic behavior. In a previous work [38], we constructed the associated harmonic measure, a key element in formulating the local limit theorem for conditioned random walks on groups. The primary aim of this paper is to prove this theorem. The main challenge arises from studying the conditioned reverse walk, whose increments, in the context of random walks on groups, depend on the entire future. To achieve our goal, we combine a Caravenna-type conditioned local limit theorem with the conditioned version of the central limit theorem for the reversed walk. The resulting local limit theorem is then applied to derive the local behavior of the exit time. |
| title | Conditioned random walks on linear groups II: local limit theorems |
| topic | Probability Dynamical Systems Primary 60B15, 60B20, 60F05, 60J05. Secondary 22D40, 22E46 |
| url | https://arxiv.org/abs/2410.05897 |