Characterization of strictly operator semi-stable distributions (Q2708264)

The author investigates some properties of strictly operator semistable distributions. These distributions possess some features of operator stable distributions on the one hand and of semistable distributions on the other. The reader can consult: \textit{M. Sharpe} [Trans. Am. Math. Soc. 136, 51-65 (1969; Zbl 0192.53603)] with respect to the notion of operator stable distribution, \textit{V. Chorny} [Theory Probab. Appl. 31, 703-705 (1987); translation from Teor. Veroyatn. Primen. 31, No. 4, 795-797 (1986; Zbl 0624.60009)] with respect to the notion of operator semistable distribution, \textit{P. Lévy} [``Théorie de l'addition des variables aléatoires'' (1954; Zbl 0056.35903)], \textit{R. Shimizu} [Ann. Inst. Stat. Math. 22, 245-255 (1970; Zbl 0236.60020)] and \textit{V. M. Kruglov} [Theory Probab. Appl. 17, 685-694 (1972); translation from Teor. Veroyatn. Primen. 17, 723-732 (1973; Zbl 0279.60035)] with respect to the notion of semistable distribution. NEWLINENEWLINENEWLINELet \(0<b<1\) and \(Q\) be a linear operator defined on the \(n\)-dimensional Euclidean space \(R^n\) whose eigenvalues have positive real parts. The infinitely divisible distribution \(\mu\) is said to be strictly \((Q,b)\)-semistable if the identity \(\mu^n=b^Q\mu\) holds, where NEWLINE\[NEWLINE b^Q=\sum_{k=0}^{\infty}{(\log b)^k\over k!} Q^k, \qquad b^Q\mu(A)=\mu(\{x\colon b^Qx\in A\}), NEWLINE\]NEWLINE \(\mu^n\) is the convolution of \(n\) copies of \(\mu\) and \(Q^k\) is the product of \(n\) copies of \(Q\). The infinitely divisible distribution \(\mu\) is said to be \((Q,b)\)-semistable if \(\mu^n\) is the convolution of \(b^Q\mu\) and the distribution \(\delta_c\) concentrated at some point \(c\in R^n\). The author pays his main attention to some characterizations of strictly \((Q,b)\)-semistable distributions and some possible relations between them and \((Q,b)\)-semistable distributions.