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An elementary proof of the Knight-Meyer characterization of the Cauchy distribution. (English) Zbl 0655.62007
This paper propounds a short proof of a result previously proved by F. Knight and P. A. Meyer [Z. Wahrscheinlichkeitstheorie verw. Gebiete 34, 129-134 (1976; Zbl 0353.60020)]. Let X be a random variable in \({\mathbb{R}}^ n\) with the following property: for any matrix \(\left( \begin{matrix} a\quad b\\ c\quad d\end{matrix} \right)\) in \(GL(n+1)\) (where a is an (n,n) matrix) there exist \(\alpha\) in GL(n) and \(\beta\) in \({\mathbb{R}}^ n\) so that \((aX+b)/(cX+d)\) and \((\alpha X+\beta)\) have the same distribution. Then X is necessarily Cauchy distributed.

MSC:
62E10 Characterization and structure theory of statistical distributions
60E05 Probability distributions: general theory
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[1] Bourbaki, N, (), Livre VI, Intégration, Chaps. 7 et 8
[2] Knight, F.B, A characterization of the Cauchy type, (), 130-135 · Zbl 0341.60009
[3] Knight, F.B; Meyer, P.A, Une caractérisation de la loi de Cauchy, Z. wahrsch. verw. gebiete, 34, 129-134, (1976) · Zbl 0353.60020
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