Semigroup stationary processes and spectral representation

Valerie Girardin and Rachid Senoussi

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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, %as symmetric, locally %stationary, stationary reducible by space transformation, %multiplicative-stationary processes or processes with independent %increments. 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.

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Bernoulli Volume 9, Number 5 (2003), 857-876.

First available in Project Euclid: 17 October 2003

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non-stationary processes positive definite functions semigroups with involution spectral representation stationary processes


Girardin, Valerie; Senoussi, Rachid. Semigroup stationary processes and spectral representation. Bernoulli 9 (2003), no. 5, 857--876. doi:10.3150/bj/1066418881.

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