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

Abstract

This 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 n with the following property: for any matrix (ca bb) in GL(n+1) (where a is a (n, n) matrix) there exist [alpha] in GL(n) and [beta] in n so that (aX + b)/(cX + d) and ([alpha]X + [beta]) have the same distribution. Then X is necessarily Cauchy distributed.Cauchy distribution characterization type projective space