CFDP 887

A New Proof of Knight's Theorem on the Cauchy Distribution

Author(s): 

Publication Date: October 1988

Pages: 6

Abstract: 

We offer a new and straightforward proof of F.B. Knight’s [3] theorem that the Cauchy type is characterized by the fact that it has no atom and is invariant under the involution i : x → –1/x. Our approach uses the representation X = tan θ where θ is uniform on (–π/2, π/2) when X is standard Cauchy. A matrix generalization of this characterization theorem is also given.

Keywords: 

Cauchy distribution, Involution, Matrix variate, Uniform distribution