Optimal Prediction of Generalized Stationary Processes

Abstract

Methods for solving optimal filtering and prediction problems for the classical stationary processes are well known since the late forties. Practice often gives rise to what is called generalized stationary processes [GV61], e.g., to white noise and to many other examples. Hence it is of interest to carry over optimal prediction and filtering methods to them. For arbitrary generalized stochastic processes this could be a challenging problem. It was shown recently [OS04] that the generalized matched filtering problem can be efficiently solved for a rather general class of S_J-generalized stationary processes introduced in [S96]. Here it is observed that the optimal prediction problem admits an efficient solution for a slightly narrower class of T_J-generalized stationary processes. Examples indicate that the latter class is wide enough to include white noise, positive frequencies white noise, as well as generalized processes occurring when the smoothing effect gives rise to a situation in which the distribution of probabilities may not exist at some time instances. One advantage of the suggested approach is that it connects solving the optimal prediction problem with inverting the corresponding integral operators S_J. The methods for the latter, e.g., those using the Gohberg-Semençul formula, can be found in the extensive literature, and we include an illustrative example where a computationally efficient solution is feasible.