Suppose $\mathbf{x} \in \mathbb{R}^{N\times 1}$ and $\mathbf{y}\in \mathbb{R}^{M\times 1}$ are two zero-mean random vectors with auto-covariances $\boldsymbol{\Sigma}_{\mathbf{x}}\in \mathbb{R}^{N\times N}$ and $\boldsymbol{\Sigma}_{\mathbf{y}}\in \mathbb{R}^{M\times M}$, and cross-covariance $\boldsymbol{\Sigma}_{\mathbf{x}\mathbf{y}}\in \mathbb{R}^{N\times M}$. Let $\mathbf{A}\in \mathbb{R}^{N\times N}$ and $\mathbf{B}\in \mathbb{R}^{M\times M}$ be square matrices.
My question is can I express $\mathbb{E}[\mathbf{x}^T\mathbf{A}\mathbf{x}\mathbf{x}^T\mathbf{A}\mathbf{x}\mathbf]$ and $\mathbb{E}[\mathbf{x}^T\mathbf{A}\mathbf{x}\mathbf{y}^T\mathbf{B}\mathbf{y}\mathbf]$ in terms of the auto-covariances, cross-covariance, and the square matrices $\mathbf{A}$, $\mathbf{B}$. I am not sure how to proceed with the problem. Any help is highly appreciated.
No. Even in the case where $N=1$ you are asking whether the fourth moment can be expressed in terms of the first and second. Obviously, the answer is NO