Journal of Applied Mathematics and Stochastic Analysis
Volume 9 (1996), Issue 3, Pages 263-270
doi:10.1155/S1048953396000251
A characterization and moving average representation for stable harmonizable processes
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.
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.