Stable and Semistable Hemigroups: Domains of Attraction and Self-Decomposability

A hemigroup of probability measures builds the set of distributions of the increments of an independent increment process. Decomposability properties of the hemigroup lead to stable respectively semistable hemigroups and enable us to show that such hemigroups appear as limits of certain functional limit theorems for operator-normed independent (non-identically) distributed random vectors. Regular variation properties of the norming operators show that the functional limit theorems are closely related to limits of infinitesimal triangular arrays of independent random vectors, i.e., to operator-self-decomposable respectively operator-semi-self-decomposable laws.