Let $\mathbf{x}$ be a multivariate Gaussian random vector with zero mean but not necessarily a diagonal covariance matrix(so that its component can be dependent). Let $\mathbf{a}$ be a deterministic vector, and $\mathbf{B}$ be a deterministic symmetric matrix.
Do we have $E[(\mathbf{a}^T\mathbf{x})^i(\mathbf{x}^T\mathbf{B}\mathbf{x})^j] = 0$ when $i$ is any odd integer and $j$ is any integer?
I am trying to see if the result that odd moments of zero-mean Gaussian being 0 generalizes to the multivariate case. In the scalar case, the previous term is just $E(aBX^{i+2j})=abE(X^{i+2j})=0$ since $i+2j$ is odd.
If we consider $E[(\mathbf{a}^T\mathbf{x})^i]$ only, since $\mathbf{a}^T\mathbf{x}$ is also a zero mean scalar Gaussian, we have that $E[(\mathbf{a}^T\mathbf{x})^i] = 0$ whenever $i$ is odd. But with the product $(\mathbf{a}^T\mathbf{x})^i(\mathbf{x}^T\mathbf{B}\mathbf{x})^j$, I do not know how it will affect the expectation.
Thank you!