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Characterization of operator-semistable distributions. (Russian. English summary) Zbl 0667.60021
The author considers symmetric characteristic functions of non-degenerate random vectors satisfying a functional equation \[ \ln f(y)=\int^{1}_{0}\ln f(c^ By)\kappa (dc),\quad y\in R^ d, \] where \(\kappa\) is a finite Borel measure on (0,1], B is a linear operator in \(R^ d\), whose eigenvalues have positive real parts and \(c^ B=\exp (B \ln c)\). Using the methods of R. Shimizu [Sankhyā, Ser. A 40, 319-322 (1978; Zbl 0422.62013)], it is shown, that such characteristic functions are operator-semistable, with an exponent \(B^*/\alpha\), where \(\alpha\) is the unique solution of the equation \(\int c^{\alpha}\kappa (dc)=1.\)
An application to the characterization of the operator-stable and operator-semistable distributions through the identical distribution of a monomial and a linear statistic is also given.
Reviewer: V.Čorný
60E07 Infinitely divisible distributions; stable distributions
62E10 Characterization and structure theory of statistical distributions
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