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.

翻译：本文针对一类依赖于小尺度参数$\varepsilon$的一维扩散过程，提出了两个极限定理：均值遍历定理和中心极限定理。这类过程在$\varepsilon \rightarrow 0$时弱收敛于均匀化扩散过程。在这些结果中，我们允许时间范围趋于无穷，即当$\varepsilon \rightarrow 0$时满足$T_\varepsilon \rightarrow \infty$。研究的新颖性源于以下情况：当$\varepsilon \rightarrow 0$时许多量是无界的，使得既有理论无法直接适用，因此需要对所有相关的$\varepsilon$依赖项进行细致分析。作为数学应用，我们进一步利用这些极限定理证明了均匀化扩散方程参数的最小距离估计量的渐近性质。