Finite-time trajectorial estimates for inhomogeneous random walks
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914153518596096 |
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| author | Ott, Sébastien Velenik, Yvan |
| author_facet | Ott, Sébastien Velenik, Yvan |
| contents | We consider integer-valued random walks with independent but not identically distributed increments, and extend to this context several classical estimates, including a local limit theorem, precise small-ball estimates (both conditional on the final point and unconditional), and bounds on the probability that the random walk trajectory remains positive up to a given time (again, both conditional on the final point and unconditional). Two key features of this work are that the bounds are non-asymptotic, holding true for finite time horizons, and, crucially, that the latter hold uniformly over an entire class of admissible increment sequences. This provides a robust framework for applications. These results are, in particular, tailored for the analysis of processes derived through a time-dependent tilting of the increments of a time-homogeneous random walk. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_09274 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite-time trajectorial estimates for inhomogeneous random walks Ott, Sébastien Velenik, Yvan Probability Statistical Mechanics Mathematical Physics We consider integer-valued random walks with independent but not identically distributed increments, and extend to this context several classical estimates, including a local limit theorem, precise small-ball estimates (both conditional on the final point and unconditional), and bounds on the probability that the random walk trajectory remains positive up to a given time (again, both conditional on the final point and unconditional). Two key features of this work are that the bounds are non-asymptotic, holding true for finite time horizons, and, crucially, that the latter hold uniformly over an entire class of admissible increment sequences. This provides a robust framework for applications. These results are, in particular, tailored for the analysis of processes derived through a time-dependent tilting of the increments of a time-homogeneous random walk. |
| title | Finite-time trajectorial estimates for inhomogeneous random walks |
| topic | Probability Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2511.09274 |