We prove a theorem on the uniform convergence of the local time of an ergodic diffusion. This result is then used to investigate certain estimators of the invariant density of an ergodic diffusion, including kernel estimators. We show that the pointwise consistency of these estimators can be strengthened to uniform consistency.

Local time and additive functionals (msc 60J55), Diffusion processes (msc 60J60), Stochastic ordinary differential equations (msc 60H10), Density estimation (msc 62G07)
CWI. Probability, Networks and Algorithms [PNA]

van Zanten, J.H. (1999). On the uniform convergence of local time and the uniform consistency of density estimators for ergodic diffusions. CWI. Probability, Networks and Algorithms [PNA]. CWI.