Limit Theorems for One-Dimensional Homogenized Diffusion Processes

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