Asymptotic normality for density kernel estimators in discrete and continuous time (Q1283848)

A central limit theorem for double arrays of \(\alpha\)-mixing sequences is established and applied to kernel density estimators for both discrete and continuous time stochastic processes. There is a remarkable difference between the two cases. It is already known from the i.i.d. case that in the first situation the full \(\sqrt n\)-consistency is not attainable. On the contrary to this for continuous time processes the authors prove a limit theorem at full rate \(\sqrt T\) under weak mixing assumptions and weak conditions imposed on the two-dimensional densities of the process.