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A characterization of the Cauchy-type distributions on boundaries of semisimple groups. (English) Zbl 0738.60006
The author proves the following theorem: Let \(G\) be a connected semisimple Lie group and \(P\) a parabolic subgroup of \(G\). Let \(K\) be a maximal compact subgroup of \(G\) and let \(m\) be the \(K\)-invariant probability measure on \(G/P\). Moreover let \(H\) be a subgroup of \(P\) such that \(HK=G\). Then for a probability measure \(\mu\) on \(G/P\) the following assertions are equivalent: (i) \(\{g\mu:g\in G\}=\{h\mu:h\in H\}\), i.e. the sets of \(G\)-types and of \(H\)-types of \(\mu\) coincide; (ii) \(\mu=hm\) for some \(h\in H\), i.e. \(\mu\) is of the same \(H\)-type as \(m\). In particular, if \(G=SL(n,\mathbb{R})\) and \(P=\{g\in G:gv\in\mathbb{R} v\}\) for some \(v\in\mathbb{R}^ n\), \(v\neq 0\), this theorem yields a characterization of the Cauchy distributions on \(\mathbb{R}^ n\) due to F. B. Knight and P. A. Meyer [Z. Wahrscheinlichkeitstheorie Verw. Geb. 34, 129-134 (1976; Zbl 0353.60020)]. Moreover, the theorem strengthens results by J.-L. Dunau and H. Senateur [J. Multivariate Anal. 22, 74-78 (1987; Zbl 0655.62007)] concerning measures on boundaries of semisimple Lie groups.

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60E05 Probability distributions: general theory
62E10 Characterization and structure theory of statistical distributions
Full Text: DOI
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