Finite de Finetti theorems in linear models and multivariate analysis

Let Xl,-.. , Xk be a sequence of random vectors. We give symmetry conditions on the joint distribution which imply that it is well approximated by a mixture of normal distributions. Examples include linear regression models, fixed and random effect analysis of variance models, and models with structured covariance matrices. The main technical tool shows that a uniformly distributed n x n orthogonal matrix has any n113 x n1/3 block well approximated (in total variation) by independent normal random variables.