A characterization and moving average representation for stable harmonizable processes. (1996) - Nikfar, M.; Soltani, A. Reza
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.