Recurrence formula for expectations of products of bilinear forms and expectations of bilinear forms and random matrices (Q1567316)

Let \(\vec X_j\) \((j=1, \dots,2n)\) denote a family of \(p\times 1\) random vectors which are normally distributed with mean zero and covariance matrix \(\langle\vec X_i\vec X_j' \rangle= \varphi_{ij}\). With this notation, a systematic way to compute the expectation of \(\prod^n_{j=1} (\vec X_{2j-1}' A_j\vec X_{2j})\) for \(A_j\) \((j=1, \dots,n)\) fixed matrices is presented. The algorithm provided involves a recurrence relation which is derived by expressing the products of the bilinear forms \(\vec X_{2j-1}'A_j \vec X_{2j}\) as a product of quadratic forms in the vectors \(\vec X_j\), the latter having been studied in an earlier work of the author.