Limit theorems for probability measures on simply connected nilpotent Lie groups
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.
different types of domains of attraction
scientific article; zbMATH DE number 4190798