Suppose \(x_ 1\), \(x_ 2\), \(x_ 3\) and \(x_ 4\) are random variables with joint Gaussian distribution and let \(a_ j=E x_ j\), \(c_{jk}=E(x_ jx_ k)\). The following fact is well-known: \[ E(x_ 1x_ 2x_ 3x_ 4)=c_{12}c_{34}+c_{13}c_{24}+c_{14}\quad c_{23}-2a_ 1a_ 2a_ 3a_ 4. \] It turns out that this relation can be used for studying the asymptotic properties of estimators, parametric and nonparametric, in stationary Gaussian processes. However there are problems in multivariate analysis and system identification where we have to deal with quantities of the type E(ABCD), i.e. with the expectations of products of random matrices whose elements have joint Gaussian distributions.
To find a suitable expression for E(ABCD) is just the problem treated in the present paper. The product ABCD is understood in the sense of Kronecker. After introducing a special operation on matrices, the authors not only derive a formula for E(ABCD), but they apply this formula to study the covariance matrix of the parameter estimator in a multivariate linear regression model.