Real gamma distribution on analytic bundles of flag varieties

This paper introduces four matrix normal distributions on analytic bundles of flag varieties, extending the separable covariance $\Phi \otimes \Psi$ with potentially variable-level ($\Psi$) and/or sample-level ($\Phi$) correlations. The joint distribution of sample variances and covariances, leading to the product moment distribution, is considered when precision matrices admit a specific form. Several well-known consequences, including the non-central Wishart distribution and normal quadratic forms, now appear as corollaries.