Strongly τ-Decomposable and Selfdecomposable Laws on Simply Connected Nilpotent Lie Groups

 Let ? be a simply connected nilpotent Lie group with Lie Algebra ? and let τ be a contraction on ?. A probability measure μ on ? is strongly τ-decomposable iff it is representable as the limit of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} for some probability ν on ?. We show that such a limit exists if and only if ν possesses a finite logarithmic moment with respect to a homogeneous norm on ?. This result is then generalized to the class of selfdecomposable laws on ?. We also show that selfdecomposable laws on ? correspond in a 1–1 way to operator selfdecomposable laws on the tangent space ?.