On convergence of kernel estimators of density with variable window width by dependent observations

In [1, 2] was studied a new type of nonparametric kernel estimators of probability density, whose window width varies depending on the sample, i.e., are data-based. These estimators were called adaptive. New estimators of density are superior in the rate of convergence to classical Rosenblatt–Parzen estimators. However, these valuable properties of estimators were obtained assuming that observations are independent. In this paper, we study properties of these adaptive estimators but assuming that the sample is realization of the stationary in the narrow sense random sequence. The simulation examples for the adaptive estimator constructed by dependent observations which is generated by autoregressive models are represented. The results of the investigation prove the advantage of the adaptive estimator over the classical Rosenblatt–Parzen estimator in the sense of the mean-square error. The rate of mean-square convergence of the limiting estimator (the so-called “ideal” estimator) to the true unknown density according to the dependent sample is found. The consistency of the adaptive estimator constructed by stationary dependent observations is proved.