Spectral generating operators for non-stationary processes
Dag Tj⊘stheim · Advances in Applied Probability · 1976
A new method for obtaining spectral-like representations for a large class of non-stationary random processes is formulated. For a wide sense stationary processX(t) in continuous-time the spectral representation is generated by a self-adjoint operatorHsuch thatX(t)=eiHtX(0). Extending certain recently established operator identities for wide sense stationary processes, it is shown that similar operators exist for classes of non-stationary processes. The representation generated by such an operator has the form and it shares some of the properties of the wide sense stationary spectral representation: it is dual in a precisely defined sense to the time domain representation ofX(t). There exists a classLof linear transformations ofX(t) such that forG∈L for some functiongdetermined byG.