A New Proof of Knight's Theorem on the Cauchy Distribution
We oﬀer a new and straightforward proof of F.B. Knight’s  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.
Phillips, Peter C.B., "A New Proof of Knight's Theorem on the Cauchy Distribution" (1988). Cowles Foundation Discussion Papers. 1131.