Limit Theorems for One-Dimensional Homogenized Diffusion Processes

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 $\epsilon$ and converge weakly to a homogenized diffusion process in the limit $\epsilon \to 0$. In these results, we allow for the time horizon to blow up such that $T_{\epsilon }\to \mathrm{\infty }$ as $\epsilon \to 0$. The novelty of the results arises from the circumstance that many quantities are unbounded for $\epsilon \to 0$, so that formerly established theory is not directly applicable here and a careful investigation of all relevant $\epsilon$-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.