Radial and directional parts of a random vector (Q1096285)

Let X be a random vector on \({\mathbb{R}}^ d\) and let \(R=| X|\) and for \(R\neq 0\) let \(W=X/R\). Necessary and sufficient conditions are given for R and W to be independent. If X has a non-singular normal distribution we show that the following three conditions are equivalent. (i) the components of X are independent and identically distributed with 0 means and positive variances. (ii) W is uniformly distributed on the unit sphere. (iii) R and W are independent.