 Cochran's theorem

Orthogonal projections of an isotropic Gaussian vector are independent. If $P$ is an orthogonal projection of rank $r$ and $Z\sim N(0,I)$, then

Cochran's theorem is a result in the field of statistics, particularly in the context of the analysis of variance (ANOVA) and the assessment of the independence of linear combinations of random variables. It is named after William G. Cochran. The theorem provides conditions under which the quadratic forms of a set of normally distributed random variables can be decomposed into independent components.