Parametric rates of nonparametric estimators and predictors for continuous time processes

Abstract

We show that local irregularity of observed sample paths provides additional information which allows nonparametric estimators and predictors for continuous time processes to reach parametric rates in mean square as well as in a.s. uniform convergence. For example, we prove that under suitable conditions the kernel density estimator $f_T$ associated with the observed sample path $(X_t, 0 \leq t \leq T)$ satisfies $$\sup_{x \epsilon \mathbb{R}}|f_T(x) - f(x)| = o(\ln_k T(\frac{\ln T}{T})^{1/2}) \quad \text{a.s.}, k \geq 1$$ where f denotes the unknown marginal density of the stationary process and where $\ln_k$ denotes the kth iterated logarithm.

The proof uses a special Borel-Cantelli lemma for continuous time processes together with a sharp large deviation inequality. Furthermore the parametric rate obtained in (1) is preserved by using a suitable sampling scheme.