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.

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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.