A Gaussian central limit theorem for trimmed products on simply connected step 2-nilpotent Lie groups
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.
scientific article; zbMATH DE number 727280
scientific article