On some classes of nonstationary parametric processes

In this paper we investigate nonstationary stochastic processes that are ch aracterized by temporal- and spectral-domain parameters with the aim of det ermining when temporal and spectral parameterizations exist simultaneously. We begin by examining the large class of purely nondeterministic nonstatio nary stochastic processes generated by passing white noise through a genera l linear time-varying filter. Then four subclasses of nonstationary paramet ric processes are studied: (1) the rational class; (2) the rational adjoint class; (3) the well-known ARMA class; and (4) the ARMA adjoint class. For each of these classes, we give membership conditions on the Green's functio n. These conditions are used to determine when minimum-order parameterizati ons are unique. Next, we use these results to give precise conditions under which a unique minimum-order process is a member of one or more of these c lasses. Although these conditions are quite restrictive, examples are inclu ded to show that these conditions do not apply to nonunique minimum-order p arameterizations. (C) 2000 The Franklin Institute. Published by Elsevier Sc ience Ltd. All rights reserved.