An elementary proof of the Knight-Meyer characterization of the Cauchy distribution

AbstractThis paper propounds a short proof of a result previously proved by F. Knight and P. A. Meyer (1976, Z. Warsch. Verw. Gebiete 34 129–134). Let X be a random variable in Rn with the following property: for any matrix (ca bb) in GL(n+1) (where a is a (n, n) matrix) there exist α in GL(n) and β in Rn so that (aX + b)(cX + d) and (αX + β) have the same distribution. Then X is necessarily Cauchy distributed