Mathematical framework for pseudo-spectra of linear stochastic difference equations

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Bujosa Brun, Andrés
García Ferrer, Antonio
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Facultad de CC Económicas y Empresariales. Instituto Complutense de Análisis Económico (ICAE)
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Although spectral analysis of stationary stochastic processes has solid mathematical foundations, this is not always so for some non-stationary cases. Here, we establish a rigorous mathematical extension of the classic Fourier spectrum to the case in which there are AR roots in the unit circle, ie, the transfer function of the linear time-invariant filter has poles on the unit circle. To achieve it we: embed the classical problem in a wider framework, the Rigged Hilbert space, extend the Discrete Time Fourier Transform and defined a new Extended Fourier Transform pair pseudo-covariance function/pseudo-spectrum. Our approach is a proper extension of the classical spectral analysis, within which the Fourier Transform pair auto-covariance function/spectrum is a particular case. Consequently spectrum and pseudo-spectrum coincide when the first one is defined.
This working paper has been accepted for publication in a future issue of IEEE Transactions on Signal Processing. Content may change prior to final publication. Citation information: DOI:10.1109/TSP.2015.2469640. 1053-587X copy right 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to See for more information.
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