When is adaptive better than optimal?

The authors attempt to predict a given stationary process using a first-order predictor whose single coefficient is adapted to the current observations using a constant gain identification algorithm. They investigate the prediction error variance as a function of the adaptation gain, i.e., the length of the memory (the number of observations) of the identification scheme. An infinite memory corresponds to the asymptotically constant optimal predictor, and a finite memory to a locally adaptive time varying predictor. It is shown that, in some specified situations, the prediction error variance associated with the finite memory adaptation scheme is smaller than the optimal variance. This can only occur if the model is specified, i.e. the structure of the optimal predictor is too simple