Invariance and independence in multivariate distribution theory

Several general results are presented whereby various properties of independence or conditional independence between certain random variables may be deduced from the symmetries enjoyed by their joint distributions. These are applied to the distributions of sample correlation and canonical correlation coefficients when the underlying data-distribution has suitable orthogonal invariance. A typical result is that, for a random sample of observations on three independent normal variables, r₁₂, r₁₃, and r_23.1 are mutually independent.