On second-order and fourth-order moments of jointly distributed random matrices: A survey
Random matrices \(Z=(X,Y)\) are considered, where the matrices \(X\) and \(Y\) have identical row numbers and normal jointly distributed entries. The paper presents second-order and fourth-order moments of \(Z\) in the tensor form in terms of expectations \(EX,\;EY\) and generalized covariance matrices \(\Omega=\{\Omega_{XX}, \Omega_{XY},\Omega_{YX},\Omega_{YY}\}\), where \(\Omega_{AB}\overset{def}{=}\;\text{cov} (\text{vec} A,\text{ vec} B),\) and each matrix \(A\) is considered as a generalized vector \(\text{vec} A =(A_{.1},\dots,A_{.n})\), where \(A_{.j}\) is the corresponding subset of entries \(\{A_{ij}\}\). The tensor relations are obtained for \(E(X t X),\;E(X' t X),\;E(X t Y)\), where the prime denotes transposing, and also for covariance matrices of \({\text{ vec}} (X t Y),\) \({\text{vec}}(X' t Y)\), \( {\text{ vec}}(X t X),\) \({\text{ vec}} (X' t X).\) Two lemmas are proved stating the identity of some of these expressions. A number of special cases is considered. The tensor expressions \(E(X'AY),\;E(XAY'),\;E(XAY),\;E(X'AY')\) are evaluated and tensor expressions are obtained for expectations and covariances of quadratic forms \(S_A=X'AX\) and \(S_B=X'BX\). As a special case, the expressions \(E(SBS)\) are evaluated for the matrices \(S\) having the Wishart distribution.
scientific article; zbMATH DE number 1563480
scientific article