Uniform convergence of curve estimators for ergodic diffusion processes

For ergodic diffusions, we consider kernel-type estimators for the invariant density, its derivatives and the drift function. Using empirical process theory for martingales, we first prove a theorem regarding the uniform weak convergence of the empirical density. This result is then used to derive uniform weak convergence for the kernel estimator of the invariant density. For kernel estimators of the derivatives of the invariant density and for a nonparametric drift estimator that was proposed by Banon, we give bounds for the rate at which the uniform distance between the estimator and the true curve vanishes. We also consider the problem of estimation from discrete-time observations. In that case, obvious estimators can be constructed by replacing Lebesgue integrals by Riemann sums. We show that these approximations are also uniformly consistent, provided that the bandwidths and the time between the observations are correctly balanced.