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

Authors

Peter C.B. Phillips

Document Type

Discussion Paper

Publication Date

10-1-1988

CFDP Number

887

CFDP 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.

Recommended Citation

Phillips, Peter C.B., "A New Proof of Knight's Theorem on the Cauchy Distribution" (1988). Cowles Foundation Discussion Papers. 1131.
https://elischolar.library.yale.edu/cowles-discussion-paper-series/1131