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

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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 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.

Suggested Citation

  • Peter C.B. Phillips, 1989. "A New Proof of Knight's Theorem on the Cauchy Distribution," Cowles Foundation Discussion Papers 887, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:887
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    File URL: https://cowles.yale.edu/sites/default/files/files/pub/d08/d0887.pdf
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