We derive limit theorems for diffusion processes that have a finite speed measure. First we prove a number of asymptotic properties of the density $rho_t = dmu_t /dmu$ of the empirical measure $mu_t$ with respect to the normalized speed measure $mu$. These results are then used to derive finite dimensional and uniform central limit theorems for integrals of the form $sqrt{tint (rho_t-1),dnu$, where $nu$ is an arbitrary finite, signed measure on the state space of the diffusion. We also consider a number of interesting special cases, such as uniform central limit theorems for Lebesgue integrals and functional weak convergence of the empirical distribution function.

Diffusion processes (msc 60J60), None of the above, but in MSC2010 section 62Fxx (msc 62F99)
CWI
CWI. Probability, Networks and Algorithms [PNA]

van Zanten, J.H. (2000). New limit theorems for regular diffusion processes with finite speed measure. CWI. Probability, Networks and Algorithms [PNA]. CWI.