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Spectral representation and asymptotic properties of certain deterministic fields with innovation components
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  • Published: 26 August 2003

Spectral representation and asymptotic properties of certain deterministic fields with innovation components

  • Guy Cohen1 &
  • Joseph M. Francos1 

Probability Theory and Related Fields volume 127, pages 277–304 (2003)Cite this article

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Abstract.

In this paper we derive the spectral and ergodic properties of a special class of homogeneous random fields, which includes an important family of evanescent random fields. Based on a derivation of the resolution of the identity for the operators generating the homogeneous field, and using the properties of measurable transformations, the spectral representation of both the field and its covariance sequence are derived. A necessary and sufficient condition for the existence of such representation is introduced. Using an analysis approach that employs the solution to the linear Diophantine equations, further characterization and modeling of the spectral properties of evanescent fields are provided by considering their spectral pseudo-density function, defined in this paper. The geometric properties of the spectral pseudo-density of the evanescent field are investigated. Finally, necessary and sufficient conditions for mean and strong ergodicity of the first and second order moments of these fields are derived. The analysis, initially carried out for complex valued random fields, is later extended to include the case of real valued fields.

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Authors and Affiliations

  1. Electrical and Computer Engineering Department, Ben-Gurion University, Beer-Sheva, 84105, Israel

    Guy Cohen & Joseph M. Francos

Authors
  1. Guy Cohen
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  2. Joseph M. Francos
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Corresponding author

Correspondence to Joseph M. Francos.

Additional information

This work was supported in part by the EU 5th Framework IHP Program, MOUMIR Project, under Grant RTN-1999-0177.

Mathematics Subject Classification (2000):62M40, 62J05

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Cohen, G., Francos, J. Spectral representation and asymptotic properties of certain deterministic fields with innovation components. Probab. Theory Relat. Fields 127, 277–304 (2003). https://doi.org/10.1007/s00440-003-0287-x

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  • Received: 06 March 2003

  • Published: 26 August 2003

  • Issue Date: October 2003

  • DOI: https://doi.org/10.1007/s00440-003-0287-x

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Key words or phrases:

  • Evanescent random fields
  • Resolution of the identity
  • Measurable transformations
  • Spectral representation
  • Linear Diophantine equations
  • Ergodicity
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