Supplements to operator-stable and operator-semistable laws on Euclidean spaces
The paper deals with the theory of operator-stable and operator- semistable laws on the Euclidean space V. Both these classes were originally defined as limit distributions but it turns out that they may be characterized by decomposability properties as well. If \(\mu\) is operator-stable, operator-semistable, respectively, then it is infinitely divisible and its Lévy spectral measure M satisfies (*) \(e^{tA}(M)=e^ tM\) for all \(t\in {\mathbb{R}}\) and some automorphism A of V, (**) \(B(M)=\beta M\) for some \(\beta\in (0,1)\) and some automorphism B of V, respectively. As is known, in the operator-stable case M can be disintegrated according to a cross-section C for the orbits \(\{e^{tA}x:t\in {\mathbb{R}}\}\), \(x\in V\). In the paper an explicit construction of a closed cross section C is performed. Furthermore, it is shown that the set of normalized Lévy spectral measures satisfying (*), (**), respectively, is a Bauer simplex; moreover, its extreme boundary is found. Finally, it is proved that the Lebesgue density of a full operator- semistable law can, in some cases, be extended to an analytic function.
operator-stable and operator-semistable laws
scientific article; zbMATH DE number 3971875
scientific article