Limit theorems for probability measures on simply connected nilpotent Lie groups
type: Item
instance of: scholarly article
title: Limit theorems for probability measures on simply connected nilpotent Lie groups
zbMATH Open document ID: 0722.60011
DOI: 10.1007/BF01258737
author: Siegfried Nobel
published in: Journal of Theoretical Probability
publication date: 1991-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 60E07
zbMATH DE Number: 4190798
zbMATH Keywords: locally compact group
zbMATH Keywords: stability of probability measures
zbMATH Keywords: different types of domains of attraction
zbMATH Keywords: nilpotent Lie group
zbMATH Keywords: continuous one-parameter semigroup
Wikidata QID: Q115393771
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01258737
OpenAlex ID: W1979353504
description: scientific article; zbMATH DE number 4190798