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International Journal of Stochastic Analysis

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International Journal of Stochastic Analysis / 1996 / Article

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Volume 9 |Article ID 324107 | https://doi.org/10.1155/S1048953396000251

M. Nikfar, A. Reza Soltani, "A characterization and moving average representation for stable harmonizable processes", International Journal of Stochastic Analysis, vol. 9, Article ID 324107, 8 pages, 1996. https://doi.org/10.1155/S1048953396000251

A characterization and moving average representation for stable harmonizable processes

M. Nikfar1 and A. Reza Soltani1

1Shiraz University, Department of Mathematics and Statistics, Center for Theoretical Physics and Mathematics, Iran

Received01 Mar 1995
Revised01 Dec 1995

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.

Copyright © 1996 Hindawi Publishing Corporation. 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.

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