On the nonparametric prediction of conditionally stationary sequences

We prove the strong consistency of estimators of the conditional distribution function and conditional expectation of a future observation of a discrete time stochastic process given a fixed number of past observations. The results apply to conditionally stationary processes (a class of processes including Markov and stationary processes) satisfying a strong mixing condition, and they extend and bring together the work of several authors in the area of nonparametric estimation. One of our goals is to provide further justification for the growing practical application of estimators in non-stationary time series and in other `non i.i.d.'~settings. Some arguments as to why such estimators should work very generally in practice, often in a nearly `optimal' way, are given. Two numerical illustrations are included, one with simulated data and the other with oceanographic data.