Conditioned random walks on linear groups I: construction of the target harmonic measure
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911736974540800 |
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| author | Grama, Ion Quint, Jean-François Xiao, Hui |
| author_facet | Grama, Ion Quint, Jean-François Xiao, Hui |
| contents | Our objective is to explore random walks on the general linear group, constrained to a specific domain, with a primary focus on establishing the conditioned local limit theorem. This paper represents the first step toward achieving this goal, specifically entailing the construction of a novel entity -- the target harmonic measure. This measure, together with the harmonic function, serves as a pivotal component in establishing the conditioned local limit theorem. Using a reversal identity, we introduce a reversed sequence characterized as a dual random walk with a perturbation depending on future observations. The investigation of such walks, which rely on future information, lies at the heart of this paper. To carry out this study, we develop an approach grounded in the finite-size approximation of perturbations, enabling us to simplify the investigation to an array of Markov chains with increasing dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05812 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conditioned random walks on linear groups I: construction of the target harmonic measure Grama, Ion Quint, Jean-François Xiao, Hui Probability Group Theory Primary 60F17, 60J05, 60J10. Secondary 22E46, 22D40 Our objective is to explore random walks on the general linear group, constrained to a specific domain, with a primary focus on establishing the conditioned local limit theorem. This paper represents the first step toward achieving this goal, specifically entailing the construction of a novel entity -- the target harmonic measure. This measure, together with the harmonic function, serves as a pivotal component in establishing the conditioned local limit theorem. Using a reversal identity, we introduce a reversed sequence characterized as a dual random walk with a perturbation depending on future observations. The investigation of such walks, which rely on future information, lies at the heart of this paper. To carry out this study, we develop an approach grounded in the finite-size approximation of perturbations, enabling us to simplify the investigation to an array of Markov chains with increasing dimensions. |
| title | Conditioned random walks on linear groups I: construction of the target harmonic measure |
| topic | Probability Group Theory Primary 60F17, 60J05, 60J10. Secondary 22E46, 22D40 |
| url | https://arxiv.org/abs/2410.05812 |