Semigroup Stationary Processes and Spectral Representation

We present an extended definition of the second-order stationarity concept. This is based on the theory of harmonic analysis for semigroups with involution. It provides a spectral representation for a wide class of processes which are non-stationary in the usual weak sense, and allows miscellaneous...

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Bibliographic Details
Published in:Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability 2003-10, Vol.9 (5), p.857-876
Main Authors: Girardin, Valerie, Senoussi, Rachid
Format: Article
Language:English
Subjects:
Covariance
Index sets
Inverted spectra
Karhunen Loeve expansion
Life Sciences
Mathematical functions
non-stationary processes
positive definite functions
Semigroups
semigroups with involution
Spectral index
spectral representation
Stationary processes
Stochastic processes
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Summary:We present an extended definition of the second-order stationarity concept. This is based on the theory of harmonic analysis for semigroups with involution. It provides a spectral representation for a wide class of processes which are non-stationary in the usual weak sense, and allows miscellaneous spectral representation results to be unified. Many applications are given to illustrate the concept. Most of these are already known, but some are new, such as the multiplicative-symmetric processes. We are less concerned with proving fundamental results than with opening up a new field of investigation for spectral representation of non-stationary processes.
ISSN:1350-7265
DOI:10.3150/bj/1066418881