Polar angle tangent vectors follow Cauchy distributions under spherical symmetry

Let X = ( X 1 , … , X n ) ′ follow a spherically or elliptically symmetric distribution centered at zero, and Y i = X i + 1 / X 1 , Y = ( Y 1 , … , Y n − 1 ) ′ . It is shown that under spherical symmetry Y has a symmetric Cauchy distribution and under elliptical symmetry a general Cauchy distribution. Geometrically, Y is the tangent (or cotangent) vector of the polar angle θ 1 . The simple case of one ratio is treated in Arnold and Brockett (1992), Jones (1999, 2008). Moreover, it is shown that n − 1 cot θ 1 follows the t n − 1 distribution, so that the normal theory distributions of Student’s t and correlation coefficient r hold under spherical symmetry.