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PEF output tends to whiteness

The most important property of a prediction-error filter or PEF is that its output tends to a white spectrum (to be proven here). No matter what the input to this filter, its output tends to whiteness as the number of the coefficients $ n \rightarrow \infty$ tends to infinity. Thus, the PE filter adapts itself to the input by absorbing all its color. This has important statistical implications and important geophysical implications.