Journal of Applied Mathematics and Stochastic Analysis
Volume 9 (1996), Issue 3, Pages 263-270

A characterization and moving average representation for stable harmonizable processes

M. Nikfar and A. Reza Soltani

Shiraz University, Department of Mathematics and Statistics, Center for Theoretical Physics and Mathematics, Tehran AEOI, Iran

Received 1 March 1995; Revised 1 December 1995

Copyright © 1996 M. Nikfar and A. Reza Soltani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In this paper we provide a characterization for symmetric α-stable harmonizable processes for 1<α2. 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<α2, then there is a function kL2(λ) (λ is the Lebesgue measure) and a certain symmetric-α-stable random measure Y for which eitxdZ(x)=k(ts)dY(s),tR, 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.