On central and non-central limit theorems in density estimation for sequences of long-range dependence
The paper studies the asymptotic properties of kernel density estimators of stationary sequences which are observed through some non-linear instantaneous filter applied to long-range dependent Gaussian sequences [see \textit{J. Beran}, Statistical methods for data with long-range dependence. Stat. Sci. 7, 404-427 (1992)]. In the three main theorems, the author shows that under long-range dependence the limiting distribution of the kernel estimator can be Gaussian or non-Gaussian depending on the choice of the bandwidth sequences. The theorems also contain some interesting properties which contrast noticeably to the short-range-dependent cases. In particular, if the bandwith \(h(N)\) for samples of size \(N\) is selected to converge to zero fast enough, the usual \(\sqrt {Nh(N)}\)-rate asymptotic normality still holds.
scientific article; zbMATH DE number 1226247
scientific article