Recursive Nonparametric Estimation for Time Series

This paper considers online kernel estimation for both short- and long-range dependent time series data. Utilizing the predictive dependence measure of Wu, we carefully study the asymptotic properties of recursive kernel density and regression estimators for a general class of stationary processes. In particular, we prove that the proposed estimators have the asymptotic normality and the corresponding central limit theorems are provided. In addition, we establish the sharp laws of the iterated logarithms that precisely characterize the asymptotic almost sure behavior of the proposed estimators.