It is clear that xt* is the projection of xt on the closed linear manifold Mt-, generated by the components of xt-, Xt-2, .... Paralleling the one-dimensional context, we call xt purely deterministic if x: = xt* and xt purely nondeterministic if the projection of on the infinite past IIMt is the null vector. It is of especial interest to be able to obtain a characterization of the class of purely nondeterministic processes in terms of the spectrum of the process. A partial result in this direction has been given by N. Wiener.' This result characterizes the class of purely nondeterministic processes with nonsingular prediction error (1). First, consider predicting xt by a linear expression in x, ..., xt-m, i.e.,
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