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

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.