Some mixing conditions for stationary symmetric stable stochastic processes
type: Item
instance of: scholarly article
title: Some mixing conditions for stationary symmetric stable stochastic processes
zbMATH Open document ID: 0813.60039
DOI: 10.1016/0304-4149(94)90046-9
author: Aaron Gross
published in: Stochastic Processes and their Applications
publication date: 1995-05-30T00:00:00Z
review text: Necessary and sufficient conditions for mixing of non-Gaussian stationary symmetric stable processes \(X_ t\), \(t \in R\), are given in terms of the spectral representation \((U,f)\), where \(U=U^ t\), \(t\in R\), is a group of isometries on the linear subspace \(L\) of some class of real-valued functions \[ L^ \alpha (E,{\mathcal E}, \mu) = \Bigl \{f : E \to R : \| f \|^ \alpha = \int | f |^ \alpha d \mu < \infty \Bigr\}, \] and a function \(f\) in \(L\), such that for any finite linear combination \(\sum a_ t X_ t\), \(a_ t\in R\), \[ E \exp \Bigl(i \sum a_ t X_ t\Bigr) = \exp \Bigl( - \Bigl\| \sum a_ t U^ tf \Bigr \|^ \alpha \Bigr). \] Two equivalence relations -- one on a collection of representations of symmetric stable processes, and the other on the collection of symmetric stable processes themselves -- are defined. It has been shown that the mixing property is well-defined on the equivalence classes. The case of stationarity of the spectral representation \((U,f)\) is also considered.
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 672051
zbMATH Keywords: weak mixing
zbMATH Keywords: stability
zbMATH Keywords: mixing of non-Gaussian stationary symmetric stable processes
zbMATH Keywords: spectral representation
MaRDI profile type: Publication
full work available at URL: 0304-4149(94)90046-9
OpenAlex ID: W2057942709
cites work: Q3794956
cites work: Ergodic properties of stationary stable processes
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cites work: On the spectral representation of symmetric stable processes
cites work: On the isometries of certain function-spaces
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description: scientific article; zbMATH DE number 672051