An ergodic theorem related to some limit theorems for additive functionals of complex Brownian motion (Q1326732)

The author proves an ergodic theorem for diffusions. We state one application: Let \(z\) be a complex Brownian motion, \(f \in L^ 1 \cap L^ p\) and \(u(t) = e^{2t} -1\). Then \(\lim_{\lambda \to \infty} \lambda^{-1/2} \int^{u(\sqrt \lambda t)}_ 0 f(z(s))ds\) converges in the sense of finite-dimensional distributions. This extends earlier results in this direction. Other applications are also given.