COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 887

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

Peter C.B. Phillips

September 1988

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 theta where theta is uniform on (-pi/2, pi/2) when X is standard Cauchy. A matrix generalization of this characterization theorem is also given.

Keywords: Cauchy distribution, Involution, Matrix variate, Uniform distribution