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The spectral representation of stable processes: Harmonizability and regularity
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  • Published: March 1990

The spectral representation of stable processes: Harmonizability and regularity

  • A. Makagon1 &
  • V. Mandrekar2 

Probability Theory and Related Fields volume 85, pages 1–11 (1990)Cite this article

  • 112 Accesses

  • 13 Citations

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Summary

We show that symmetric α-stable moving average processes are not harmonizable. However, we show that a concept of generalized spectrum holds for allL p -bounded processes O<p<-2. In capep=2, generalized spectrum is a measure and the classical representation follows. For strongly harmonizable symmetric α-stable processes we derive necessary and sufficient conditions for the regularity and the singularity for 0<α≦2, using known results on the invariant subspaces. We also get Cramér-Wold decomposition for the case 0<α≦2.

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Author information

Authors and Affiliations

  1. Institute of Mathematics, Wroclaw Technical University, 50-370, Wroclaw, Poland

    A. Makagon

  2. Department of Statistics and Probability, Michigan State University, Wells Hall, 48824-1027, East Lansing, MI, USA

    V. Mandrekar

Authors
  1. A. Makagon
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  2. V. Mandrekar
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Additional information

Supported by CPBP01. 02.

Supported by ONR N00014-85-K-0150

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Cite this article

Makagon, A., Mandrekar, V. The spectral representation of stable processes: Harmonizability and regularity. Probab. Th. Rel. Fields 85, 1–11 (1990). https://doi.org/10.1007/BF01377623

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  • Received: 21 March 1988

  • Revised: 10 October 1989

  • Issue Date: March 1990

  • DOI: https://doi.org/10.1007/BF01377623

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Keywords

  • Stochastic Process
  • Probability Theory
  • Classical Representation
  • Mathematical Biology
  • Average Process
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