Some mixing conditions for stationary symmetric stable stochastic processes
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.
mixing of non-Gaussian stationary symmetric stable processes
scientific article; zbMATH DE number 672051
scientific article