Expectation of cubic form (covariance between a quadratic form and a dot product)

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 $$\mathrm{Cov}(X^T A X, b^T X)=\mathbb{E}[ (X^TAX) (b^TX)] \text{ ?}$$

• This will involve third order moments of components of $X$, which you cannot obtain from only the mean and covariance matrix. – angryavian Feb 14 at 16:54
• Agreed. Assuming that you know everything about the 3rd order moments of $X$, what can be said? – alezok Feb 14 at 20:46
• 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 $E[x_i x_j x_k]$. – angryavian Feb 14 at 20:55