Strictly operator-stable distributions (Q1095486)

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.