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A characterization and moving average representation for stable harmonizable processes

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  • M. Nikfar
  • A. Reza Soltani

Abstract

In this paper we provide a characterization for symmetric α -stable harmonizable processes for 1 < α ≤ 2 . We also deal with the problem of obtaining a moving average representation for stable harmonizable processes discussed by Cambanis and Soltani [3], Makegan and Mandrekar [9], and Cambanis and Houdre [2]. More precisely, we prove that if Z is an independently scattered countable additive set function on the Borel field with values in a Banach space of jointly symmetric α -stable random variables, 1 < α ≤ 2 , then there is a function k ∈ L 2 ( λ ) ( λ is the Lebesgue measure) and a certain symmetric- α -stable random measure Y for which ∫ − ∞ ∞ e i t x d Z ( x ) = ∫ − ∞ ∞ k ( t − s ) d Y ( s ) , t ∈ R , if and only if Z ( A ) = 0 whenever λ ( A ) = 0 . Our method is to view S α S processes with parameter space R as S α S processes whose parameter spaces are certain L β spaces.

Suggested Citation

  • M. Nikfar & A. Reza Soltani, 1996. "A characterization and moving average representation for stable harmonizable processes," International Journal of Stochastic Analysis, Hindawi, vol. 9, pages 1-8, January.
  • Handle: RePEc:hin:jnijsa:324107
    DOI: 10.1155/S1048953396000251
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