Semistable distributions in \({\mathbb{R}}^ d\) (Q2266281)

Let \(\kappa\) be an \(\sigma\)-finite measure defined on a \(\sigma\)-algebra of Borel sets of the segment [-1,1]. A probability distribution \(\mu\) in \(R^ d\) will be called semistable if its characteristic function does not vanish and satisfies the equation \(\ln{\hat\mu}(t) = \int^{1}_{- 1} \ln{\hat\mu}(ct) \kappa(dc)\). It is shown that if \({\hat\mu}(t)\) is a semistable characteristic function and its measure \(\kappa\) satisfies the following conditions: Supp(\(\kappa)\not\subset \{-1,1\}\), \(\kappa(\{- 1,1\})\), \(\kappa([-1,1])>1\), then \({\hat\mu}(t)\) is infinitely divisible. The proof can be carried out by a method similar to that employed in \textit{A. M. Kagan, Yu. V. Linnik} and \textit{S. R. Rao}, Characterization problems of mathematical statistics. 228-264 (1972; Zbl 0243.62009). Furthermore a complete characterization of the semistable distribution is offered in terms of the canonical Lévy representation. Finally it is also pointed out that semistable distributions are absolutely continuous.