A Clarification of the Cauchy Distribution

Abstract WedefineamultivariateCauchydistributionusingaprobabilitydensityfunction; subsequently, aFerguson’sdefinition of a multivariate Cauchy distribution can be viewed as a characterization theorem using the character-istic function approach. To clarify this characterization theorem, we construct two dependent Cauchy randomvariables, but their sum is not Cauchy distributed. In doing so the proofs depend on the characteristic function,but we use the cumulative distribution function to obtain the exact density of their sum. The derivation methodsare relatively straightforward and appropriate for graduate level statistics theory courses.Keywords: Cauchy distribution, dependency, linear combination, characteristic function, distribu-tion function. 1. Introduction The Cauchy distribution is an example of a distribution which has no mean, variance or higher mo-ments defined. Hence it has no moment generating function (mgf). The sample mean will have thesame standard Cauchy distribution if X 1