The prediction error of stationary Gaussian time series of unknown covariance

In prediction problems of communication and control theory, it has become increasingly obvious that there are many applications in which a priori assumptions regarding data statistics are not justified. Thus, systems must be designed to take maximum advantage of whatever statistics are encountered. Unfortunately, these systems are inherently nonlinear in operation, which makes it difficult, ff not impossible, to evaluate their performance. In this paper the asymptotic form of the mean square prediction error is found for a stationary Gaussian time series when the prediction is a linear weighting of the immediate past, the weights being "learned" from the data. Computer results are given to demonstrate the usefulness of the asymptotic formula.