Abstract: We consider the random-design nonparametric regression model with errors an unknown function of long-range dependent moving averages and of the independent and identically distributed explanatory random vectors. We show that the Nadaraya-Watson kernel estimator of the regression function may exhibit a dichotomous asymptotic behavior depending on the amount of smoothing employed: its finite-dimensional distributions converge either to those of a not necessarily Gaussian degenerate process with completely dependent marginals or to those of a Gaussian white noise process. The borderline situation results in a limiting convolution of the two cases. Convergence to Gaussian white noise is also established when the resulting errors lack a long memory. The main results here are general analogues of those in Csörgo and Mielniczuk (1999), where the smoothing dichotomy was disclosed in the case when the long-range dependence of the errors is produced by a Gaussian sequence.
Key words and phrases: Finite-dimensional distributions, kernel estimators, long-memory errors, moving averages, random-design nonparametric regression, smoothing dichotomy.