The Wiener Theory of Causal Linear Prediction Is Not Effective

In this paper, it will be shown that the minimum mean square error (MMSE) for predicting a stationary stochas-tic time series from its past observations is not generally Turing computable, even if the spectral density of the stochastic process is differentiable with a computable first derivative. This implies that for any approximation sequence that converges to the MMSE there does not exist an algorithmic stopping criterion that guarantees that the computed approximation is sufficiently close to the true value of the MMSE. Furthermore, it will be shown that under the same conditions on the spectral density, it is also the case that coefficients of the optimal prediction filter are not generally Turing computable.