Invariance and independence in multivariate distribution theory (Q1082009)

In a variety of cases, symmetries in a multivariate distribution can be used to demonstrate independence between certain functions of random variables. Several relevant theorems are collected from a variety of sources, and used to demonstrate independence in several different situations. For example, if a random sample is taken on three independent normal variables, the two sample correlation coefficients \(r_{12}\) and \(r_{13}\), and the sample partial correlation coefficient \(r_{23.1}\) are mutually independent.