On some classes of nonstationary parametric processes

In this paper we investigate nonstationary stochastic processes that are characterized by temporal- and spectral-domain parameters with the aim of determining when temporal and spectral parameterizations exist simultaneously. We begin by examining the large class of purely nondeterministic nonstationary stochastic processes generated by passing white noise through a general linear time-varying filter. Then four subclasses of nonstationary parametric 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 Greenʹs function. These conditions are used to determine when minimum-order parameterizations 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 classes. Although these conditions are quite restrictive, examples are included to show that these conditions do not apply to nonunique minimum-order parameterizations.