Limit Theorems for One-Dimensional Homogenized Diffusion Processes
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908606105911296 |
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| author | Borodavka, Jaroslav I. Krumscheid, Sebastian |
| author_facet | Borodavka, Jaroslav I. Krumscheid, Sebastian |
| contents | We present two limit theorems, a mean ergodic and a central limit theorem, for a specific class of one-dimensional diffusion processes that depend on a small-scale parameter $\varepsilon$ and converge weakly to a homogenized diffusion process in the limit $\varepsilon \rightarrow 0$. In these results, we allow for the time horizon to blow up such that $T_\varepsilon \rightarrow \infty$ as $\varepsilon \rightarrow 0$. The novelty of the results arises from the circumstance that many quantities are unbounded for $\varepsilon \rightarrow 0$, so that formerly established theory is not directly applicable here and a careful investigation of all relevant $\varepsilon$-dependent terms is required. As a mathematical application, we then use these limit theorems to prove asymptotic properties of a minimum distance estimator for parameters in a homogenized diffusion equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06691 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limit Theorems for One-Dimensional Homogenized Diffusion Processes Borodavka, Jaroslav I. Krumscheid, Sebastian Probability Statistics Theory We present two limit theorems, a mean ergodic and a central limit theorem, for a specific class of one-dimensional diffusion processes that depend on a small-scale parameter $\varepsilon$ and converge weakly to a homogenized diffusion process in the limit $\varepsilon \rightarrow 0$. In these results, we allow for the time horizon to blow up such that $T_\varepsilon \rightarrow \infty$ as $\varepsilon \rightarrow 0$. The novelty of the results arises from the circumstance that many quantities are unbounded for $\varepsilon \rightarrow 0$, so that formerly established theory is not directly applicable here and a careful investigation of all relevant $\varepsilon$-dependent terms is required. As a mathematical application, we then use these limit theorems to prove asymptotic properties of a minimum distance estimator for parameters in a homogenized diffusion equation. |
| title | Limit Theorems for One-Dimensional Homogenized Diffusion Processes |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2503.06691 |