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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.

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