On Series Representations for Linear Predictors

Abstract

The series expressions for the linear predictors of a stationary process have been known for a long time, but necessary and sufficient conditions for the mean square convergence of these series are still not available. It is shown that an equivalent problem is to find necessary and sufficient conditions for the invertibility of the infinite moving average representation of the process. Two known sufficient conditions are discussed, and a more general condition that includes both as special cases is given. The process that arises from fractional differencing of a random walk is discussed as an example.