Strictly operator-stable distributions
type: Item
instance of: scholarly article
title: Strictly operator-stable distributions
zbMATH Open document ID: 0632.60013
DOI: 10.1016/0047-259X(87)90091-1
author: Ken-Iti Sato
published in: Journal of Multivariate Analysis
publication date: 1987-01-01T00:00:00Z
review text: A process X(t), \(t\geq 0\), on \({\mathbb{R}}^ d\) is called operator stable if X(t)\(=^{d}T_ tX(1)+a_ t\), where \(T_ t\), \(t\geq 0\), are invertible linear operators on \({\mathbb{R}}^ d\) and \(a_ t\in {\mathbb{R}}^ d\); it is called strictly operator stable if \(a_ t\) can be taken zero. X(t) is called (strictly) (\(\alpha\),Q)-stable if \(X(t^{\alpha})=^{d}t^ QX(1)+a_ t\) (with \(a_ t=0).\) Extending work by \textit{M. Sharpe} [Trans. Am. Math. Soc. 136, 51-65 (1969; Zbl 0192.536)], \textit{W. N. Hudson} and \textit{J. D. Mason} [J. Multivariate Anal. 11, 434-447 (1981; Zbl 0466.60016)] and the author and \textit{M. Yamazato} [Nagoya Math. J. 97, 71-94 (1985; Zbl 0577.60025)], the author characterizes the (\(\alpha\),Q)-stable distributions that are strictly (\(\alpha\),Q)-stable, for \(\alpha\) an eigenvalue of Q. The characterization is in terms of conditions on the Lévy representation for (\(\alpha\),Q)-stable distributions as obtained earlier by the above- mentioned authors. The case at hand differs from the previous ones by the fact that no translation of an (\(\alpha\),Q)-stable distribution may be strictly stable. Some special cases are given and the (larger) class of limits of the form d-\(\lim_{n\to \infty}T_ n(X_ 1+...+X_ n)\) is considered briefly.
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60E10
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 4028499
zbMATH Keywords: characteristic function
zbMATH Keywords: linear transformation
zbMATH Keywords: operator stable
zbMATH Keywords: strictly operator stable
zbMATH Keywords: eigenvalue
zbMATH Keywords: Lévy representation
reviewed by: Fred W. Steutel
MaRDI profile type: Publication
cites work: Q3671491
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description: scientific article; zbMATH DE number 4028499