Nonparametric density and regression estimation for Markov sequences without mixing assumptions

The nonparametric estimation results for time series described in the literature to date stem fairly directly from a seminal work of M. Rosenblatt. The gist of the current picture is that under either strong or G2 mixing, many properties of nonparametric estimation in the i.i.d. case carry over to Markov sequences as well. The present work shows that many of the above results remain valid even when mixing assumptions are removed altogether. It is seen here that if the Markov process has a stationary density function, then under standard smoothness conditions, the kernel estimators of the stationary density and the auto-regression functions are asymptotically normal, with the same limiting parameters as in the i.i.d. case. Even when no stationary law exists, there are circumstances lenient enough to include ARMA processes and random walks, for which a kernel auto-regression estimator with sample-driven bandwidths is asymptotically normal. The foundation for this study is developments by Orey and Harris.