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Spectral generating operators for non-stationary processes

Published online by Cambridge University Press:  01 July 2016

D. Tj⊘stheim
D. Tj⊘stheim*
Affiliation:
Royal Norwegian Council for Scientific and Industrial Research (NORSAR)
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Abstract

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.

Keywords

NON-STATIONARY PROCESSESSPECTRAL GENERATING OPERATORSCOMMUTATION RELATIONSTIME-FREQUENCY DUALISMINNOVATIONS STABILITYTIME-VARYING SPECTRAL DENSITYSPECTRAL ANALYSERS

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 4 , December 1976 , pp. 831 - 846
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Akhiezer, N. I. and Glazman, I. M. (1963) Theory of Linear Operators in Hilbert Space. Vol. 2. Ungar, New York.Google Scholar
[2] Cramér, H. (1961) On the structure of purely non-deterministic stochastic processes. Ark. Mat. 4, 249–266.CrossRefGoogle Scholar
[3] Cramér, H. (1961) On some classes of non-stationary stochastic processes. Proc. 4th Berkeley Symp. Math. Statist. Prob. 2, 55–77.Google Scholar
[4] Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[5] Hanner, O. (1950) Deterministic and non-deterministic processes. Ark. Mat. 1, 161–177.CrossRefGoogle Scholar
[6] Kallianpur, G. and Mandrekar, V. (1965) Multiplicity and representation theory of purely non-deterministic stochastic processes. Theor. Prob. Appl. 10, 553–581.Google Scholar
[7] Loynes, R. M. (1968) On the concept of the spectrum for non-stationary processes. J. R. Statist. Soc. B 30, 1–30.Google Scholar
[8] Von Neumann, J. (1931) Die Eindeutigkeit der Schröderingerschen Operatoren. Math. Ann. 104, 570–578.CrossRefGoogle Scholar
[9] Priestley, M. B. (1965) Evolutionary spectra and non-stationary processes. J. R. Statist. Soc. B 27, 204–229.Google Scholar
[10] Priestley, M. B. and Tong, H. (1973) On the analysis of bivariate non-stationary processes. J. R. Statist. Soc. B 35, 153–166.Google Scholar
[11] Putnam, C. R. (1967) Commutation Properties of Hilbert Space Operators and Related Topics. Springer, New York.Google Scholar
[12] Riesz, F. and Nagy, Sz., de, B. (1955) Functional Analysis, 2nd edn. Ungar, New York.Google Scholar
[13] Tj⊘stheim, D. (1976) A commutation relation for wide sense stationary processes. SIAM J. Appl. Math. 30, 115–122.Google Scholar
[14] Tj⊘stheim, D. (1976) Multiplicity theory for multivariate wide sense stationary generalized processes. J. Multivariate Anal. 5, 314–321.CrossRefGoogle Scholar
[15] Whittle, P. (1968) Contribution to the discussion of R. M. Loynes ‘On the concept of the spectrum for a non-stationary process.’ J. R. Statist. Soc. B 30, 26.Google Scholar