Strongly τ-decomposable and selfdecomposable laws on simply connected nilpotent Lie groups

Let G be a simply connected nilpotent Lie group with Lie Algebra V and let tau be a contraction on G. A probability measure mu on G is strongly tau-decomposable iff it is representable as the limit of nu*...tau(n)(nu) for some probability nu on G. We show that such a limit exists if and only if nu possesses a finite logarithmic moment with respect to a homogeneous norm on G. This result is then generalized to the class of selfdecomposable laws on G. We also show that selfdecomposable laws on G correspond in a 1-1 way to operator selfdecomposable laws on the tangent space V.