Polar angle tangent vectors follow Cauchy distributions under spherical symmetry

Let X=(X1,…,Xn)′ follow a spherically or elliptically symmetric distribution centered at zero, and Yi=Xi+1/X1, Y=(Y1,…,Yn−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 θ1. The simple case of one ratio is treated in Arnold and Brockett (1992), Jones (1999, 2008). Moreover, it is shown that n−1cotθ1 follows the tn−1 distribution, so that the normal theory distributions of Student’s t and correlation coefficient r hold under spherical symmetry.