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Consider a $n$-dimensional random vector $X$ with mean $\mathbb{E}X=\mathbf{0}$ and covariance matrix $\Sigma$. Given a $n\times n$ matrix $A$ and a $n$-dimensional vector $b$, is there any explicit expression for the following quantity $$ \mathrm{Cov}(X^T A X, b^T X)=\mathbb{E}[ (X^TAX) (b^TX)] \text{ ?} $$

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    $\begingroup$ This will involve third order moments of components of $X$, which you cannot obtain from only the mean and covariance matrix. $\endgroup$ – angryavian Feb 14 at 16:54
  • $\begingroup$ Agreed. Assuming that you know everything about the 3rd order moments of $X$, what can be said? $\endgroup$ – alezok Feb 14 at 20:46
  • $\begingroup$ You could write out the matrix multiplication as $(\sum_i \sum_j A_{ij} x_i x_j)(\sum_k b_k x_k)$, multiply through, collect terms, and use whatever information you have about $E[x_i x_j x_k]$. $\endgroup$ – angryavian Feb 14 at 20:55

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