Limit theorems for probability measures on simply connected nilpotent Lie groups (Q756234)

Let G be an aperiodic, strongly root-compact, second countable, locally compact group or, more specifically, a simply connected nilpotent Lie group. The aim of the article is to compare different types of stability of probability measures and convolution semigroups on G with different types of domains of attraction. As a main result, it is shown that for a full probability measure \(\mu\) on a simply connected nilpotent Lie group G the following assertions are equivalent: (a) There exist \(\nu \in M^ 1(G)\) and \(\tau_ n\in Aut(G)\) (n\(\in {\mathbb{N}})\) such that \(\tau_ n(\nu^ n)\) tends weakly to \(\mu\). (b) There exist a continuous convolution semigroup \((\mu_ t)_{t\geq 0}\) on G and a continuous one-parameter semigroup \((\tau_ t)_{t>0}\subset Aut(G)\) such that \(\mu_ 1=\mu\) and \(\tau_ s(\mu_ t)=\mu_{st}\) for \(s,t>0.\) (c) For each \(n\in {\mathbb{N}}\) there exists \(\tau_ n\in Aut(G)\) satisfying \(\tau_ n(\mu)=\mu^ n.\) Some variants of these results are also discussed where some conclusions remain true under weaker assumptions. For instance, \(\mu\) being full may be dropped sometimes.