Ergodic properties of stationary stable processes
type: Item
instance of: scholarly article
title: Ergodic properties of stationary stable processes
zbMATH Open document ID: 0612.60034
DOI: 10.1016/0304-4149(87)90024-X
author: Stamatis Cambanis
author: Clyde D. jun. Hardin
author: Aleksander Weron
published in: Stochastic Processes and their Applications
publication date: 1987-01-01T00:00:00Z
review text: A real random variable Y is called symmetric \(\alpha\)-stable (S\(\alpha\) S) for some \(\alpha\in (0,2]\), if E exp(irY)\(=\exp (-c_ Y| r|^{\alpha})\) for all r and some \(c_ Y\geq 0\). A process \(\{X_ t\}\) is called \(S\alpha\) S if all finite linear combinations \(\sum a_ jX_{t_ j}\) are \(S\alpha\) S. The authors derive conditions for stationary \(S\alpha\) S processes to be metrically transitive and mixing. Sub-Gaussian stationary processes and \(S\alpha\) S processes with a harmonic spectral representation are never metrically transitive. However, the latter satisfy a strong law of large numbers (SLLN) even though they are not generally stationary. Doubly stationary \(S\alpha\) S processes are defined in the paper and conditions for their metric transitivity, mixing and SLLN are given.
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 28D10
zbMATH DE Number: 3990511
zbMATH Keywords: stable processes
zbMATH Keywords: ergodic theory
zbMATH Keywords: metrically transitive and mixing
zbMATH Keywords: harmonic spectral representation
zbMATH Keywords: strong law of large numbers
reviewed by: Jiří Anděl
MaRDI profile type: Publication
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full work available at URL: 0304-4149(87)90024-x
OpenAlex ID: W1977915641
description: scientific article; zbMATH DE number 3990511