A Gaussian central limit theorem for trimmed products on simply connected step 2-nilpotent Lie groups
type: Item
instance of: scholarly article
title: A Gaussian central limit theorem for trimmed products on simply connected step 2-nilpotent Lie groups
zbMATH Open document ID: 0820.60003
DOI: 10.1007/BF02213459
author: Daniel Neuenschwander
published in: Journal of Theoretical Probability
publication date: 1995-02-26T00:00:00Z
review text: The aim of this paper is to prove for simply connected step-two-nilpotent groups \(G\) (hence non-Abelian groups) an analogon to a CLT of \textit{J. Kuelbs} and \textit{M. Ledoux} [Probab. Theory Relat. Fields 74, 341-355 (1986; Zbl 0586.60017)] for trimmed sums on \(G\), and let \(\nu\) belong to the domain of attraction of \((\mu_ t)\). More precisely, let \((\tau_ t = t^ A)\) be a continuous one-parameter group of automorphisms such that \(\tau_ t \mu_ 1 = \mu_ t\), \(t> 0\), and let for some sequence \((t_ n > 0, x_ n \in G)\), \(\tau_{t_ n}(\nu * \varepsilon_{x_ n})^ n \to \mu_ 1 = \mu\). \((\tau_ t)\) defines a polar decomposition of \(G\), hence random variables \(X\) are representable as \(X = T^ A Y\), where \(\tau > 0\) and \(Y\) takes its values in a suitable cross-section. Let \(X_ n\) be i.i.d. r.v. with distribution \(\nu\), let \(T_ n\) be the radial part of the polar decomposition. Let \(r_ n \to \infty\), \(r_ n/n \to 0\). Then, choosing a trimming procedure for the radial parts \(T_ n\) (similar to the procedure in the above mentioned paper of Kuelbs and Ledoux), the author proves that the trimmed product, normalized by \(r_ n^{-1/2}\), converges weakly to a centered Gauss measure. The proof is based on the usual tools for limit theorems on exponential Lie groups which allow to reduce proofs of convergence of probabilities on the group to the corresponding assertions for the tangent space. But, Lemma 2.3 depends essentially on the step-2-property of the underlying group. Thus the proof only works on Heisenberg-like groups. There is no comparable result for more general classes of groups.
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 727280
zbMATH Keywords: Heisenberg group
zbMATH Keywords: nilpotence
zbMATH Keywords: stable convolution semigroup
zbMATH Keywords: polar decomposition
zbMATH Keywords: trimmed product
zbMATH Keywords: exponential Lie groups
Wikidata QID: Q115392709
reviewed by: Wilfried Hazod
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 727280