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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:
 6.2e+11 Characterization and structure theory of statistical distributions 6e+06 Probability distributions: general theory
##### Keywords:
Cauchy distribution; characterization; type; projective space; GL(n)
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##### References:
 [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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