A Clarification of the Cauchy Distribution

Hwi-Young Lee, Hyoung-Jin Park, Hyoung-Moon Kim
2014 Communications for Statistical Applications and Methods  
We define a multivariate Cauchy distribution using a probability density function; subsequently, a Ferguson's definition of a multivariate Cauchy distribution can be viewed as a characterization theorem using the characteristic function approach. To clarify this characterization theorem, we construct two dependent Cauchy random variables, 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
more » ... on function to obtain the exact density of their sum. The derivation methods are relatively straightforward and appropriate for graduate level statistics theory courses.
doi:10.5351/csam.2014.21.2.183 fatcat:tcycen3tbzdtlnex5x6vtopahy