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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?


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What is known about these rv's? Do they have a joint denisty function? Are they jointly normal? Also, in view of the notation and the tag, are these rv's multivariate normal? – Shai Covo Dec 12 '10 at 13:12
Indeed, I neglected to mention they are both jointly normal and multivariate normal. – asd123 Dec 12 '10 at 14:34
Isn't this just more or less the matrix of covariances between the random variables in $x_1$ and the random variables in $x_2$? – Deane Yang Mar 26 '11 at 0:54

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!

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