A characterization and moving average representation for stable harmonizable processes

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∈L2(λ) (λ is the Lebesgue measure) and a certain symmetric-α-stable random measure Y for which ∫−∞∞eitxdZ(x)=∫−∞∞k(t−s)dY(s),t∈R, if and only if Z(A)=0 whenever λ(A)=0. Our method is to view SαS processes with parameter space R as SαS processes whose parameter spaces are certain Lβ spaces.