An elementary proof of the Knight-Meyer characterization of the Cauchy distribution (Q1109443)

This paper propounds a short proof of a result previously proved by \textit{F. Knight} and \textit{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.