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 non-parametric estimation. One of our goals is to provide further justification for the growing practical application of non-parametric 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.