Random integral representations for classes of limit distributions similar to Lévy class \(L_ 0\) (Q1093232)

For a bounded linear operator Q, on a Banach space E, and a real number \(\beta\), there are introduced classes, \({\mathcal U}_{\beta}(Q)\), of some limit distributions such that \({\mathcal U}_ 0(I)\) coincides with the Lévy class \(L_ 0\). Elements from \({\mathcal U}_{\beta}(Q)\) are characterized in terms of convolution equations and as probability distributions of some random integral functionals. The continuity and fixed points of this random mapping are studied. It is shown that fixed points coincide with the class of Q-stable measures.