Integrated consistency of smoothed probability density estimators for stationary sequences

Estimators of the form of the (marginal) probability density for a strictly stationary of order 2 sequence are considered. This class of estimators includes the kernel type. The properties of such estimators are discussed on the basis of their mean integrated square error (MISE). Integrated consistency results are obtained for two classes of probability densities. Many of the definitions and results are analogous to those of Parzen for the spectral density and Watson and Leadbetter for the probability density of an i.i.d. sample.