On central and non-central limit theorems in density estimation for sequences of long-range dependence

This paper studies the asymptotic properties of the kernel probability density estimate of stationary sequences which are observed through some non-linear instantaneous filter applied to long-range dependent Gaussian sequences. It is shown that the limiting distribution of the kernel estimator can be, in quite contrast to the case of short-range dependence, Gaussian or non-Gaussian depending on the choice of the bandwidth sequences. In particular, if the bandwidth h(N) for sample of size N is selected to converge to zero fast enough, the usual √Nh(N) rate asymptotic normality still holds.