Large Deviation Principle for the Empirical Measures of Simple Random Walks on $\overline{\mathbb{Z}}$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917536322289664 |
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| author | Fatras, Jan-Luka |
| author_facet | Fatras, Jan-Luka |
| contents | In this article we establish a large deviation principle for the empirical measures of a simple spatially inhomogeneous random walk on $\overline{\mathbb{Z}}$, the two-point compactification of $\mathbb{Z}$. The classical Donsker--Varadhan framework does not apply, since the random-walk kernel and the topology of $\overline{\mathbb{Z}}$ fall outside its standard assumptions. In certain regimes, the resulting rate function is non-convex on its effective domain. We also derive a large deviation principle for empirical means of observables $f:\mathbb{Z} \to \mathbb{R}^d$ admitting limits at $\pm\infty$. This result is optimal in the sense that in general, no large deviation principle holds for the larger class of bounded continuous functions on $\mathbb{Z}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_26804 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Large Deviation Principle for the Empirical Measures of Simple Random Walks on $\overline{\mathbb{Z}}$ Fatras, Jan-Luka Probability 60F10, 60G50, 60J10 In this article we establish a large deviation principle for the empirical measures of a simple spatially inhomogeneous random walk on $\overline{\mathbb{Z}}$, the two-point compactification of $\mathbb{Z}$. The classical Donsker--Varadhan framework does not apply, since the random-walk kernel and the topology of $\overline{\mathbb{Z}}$ fall outside its standard assumptions. In certain regimes, the resulting rate function is non-convex on its effective domain. We also derive a large deviation principle for empirical means of observables $f:\mathbb{Z} \to \mathbb{R}^d$ admitting limits at $\pm\infty$. This result is optimal in the sense that in general, no large deviation principle holds for the larger class of bounded continuous functions on $\mathbb{Z}$. |
| title | Large Deviation Principle for the Empirical Measures of Simple Random Walks on $\overline{\mathbb{Z}}$ |
| topic | Probability 60F10, 60G50, 60J10 |
| url | https://arxiv.org/abs/2605.26804 |