The predictability of continuous-time, bandlimited processes

In communications and signal processing, we can find examples of applications that could benefit from the prediction of a bandlimited random process. We consider a continuous-time linear predictor applied to a bandlimited process. We show that if the past values of the process are known over an interval of arbitrary positive length, the mean squared prediction error may be made arbitrarily small, regardless of how far in the future we wish to make the prediction. We also show that this is no longer true when a certain energy constraint is applied to the predictor. Furthermore, we discuss what this means for the case in which the prediction is based on past values that are corrupted by estimation errors.