Multivariate distributions with Gaussian conditional structure

Multivariate distributions exhibiting some features of the conditional structure associated with the classical normal model are investigated. Features considered include conditional distributions of subvectors and conditional moments. Our understanding of the classical normal model is enhanced by the study of such quasi-Gaussian distributions together with investigation of additional assumptions required to characterize the classical normal model. Special attention is paid to the class of distributions exhibiting Gaussian conditional structure of the second order, i.e. those in which the conditional moments of orders one and two match the Gaussian model.