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.