Thorin classes of Lévy processes and their transforms (Q616540)

This paper deals with Lévy processes in \(\mathbb{R}^d\) and their transforms under subordination. Investigating the infinite divisibility of Pareto an lognormal distributions O.Thorin introduced the important class of generalized Gamma convolutions as the minimal class of probability distributions containing all Gamma distributions, closed under convolutions and weak convergence. The author defines and characterizes Thorin classes \(T^{(\kappa)}(\mathbb{R}_+)\), \(\kappa>0\), if infinitely divisible distributions on \(\mathbb{R}_+\). He investigates Poisson, Karlin and Bessel transforms of Thorin classes and considers extended Thorin classes \(T^{(\kappa)}(\mathbb{R}^d)\), \(\kappa>0\). Canonical representation and self-decomposability properties of Thorin subordinated Gaussian Lévy processes are discussed. Finally as an example, a subordinated Cauchy processes is considered.