Say we have two multivariate Gaussian random vectors $p(x_1) = N(0,\Sigma_1), p(x_2) = N(0,\Sigma_2)$, is there a well known result for the expectation of their product $E[x_1x_2^T]$ (matrix result) without assuming independence?
Thanks.
Say we have two multivariate Gaussian random vectors $p(x_1) = N(0,\Sigma_1), p(x_2) = N(0,\Sigma_2)$, is there a well known result for the expectation of their product $E[x_1x_2^T]$ (matrix result) without assuming independence? Thanks. 


Without assuming anything about the random vectors you can only get upper and lower bounds. For a complete description of the matrix $$ \mathbb{E}[x_1x_2^{T}] $$ you need to know the correlation between the $i$th entry of the vector $x_1$ and the $j$th entry of the vector $x_2$ for all possibles indexes $i$ and $j$. I hope it helps! 

