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Spectral generating operators for non-stationary processes
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- Journal:
- Advances in Applied Probability / Volume 8 / Issue 4 / December 1976
- Published online by Cambridge University Press:
- 01 July 2016, pp. 831-846
- Print publication:
- December 1976
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- Article
- Export citation
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A new method for obtaining spectral-like representations for a large class of non-stationary random processes is formulated. For a wide sense stationary process X(t) in continuous-time the spectral representation is generated by a self-adjoint operator H such that X(t)= eiHt X(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 of X(t). There exists a class L of linear transformations of X(t) such that for G ∈ L
for some function g determined by G.
