Let $\mathbf{x}$ and $\mathbf{y}$ be Gaussian random vectors with zero mean and covariances $\mathbf{C}_{\mathbf{xx}}$, $\mathbf{C}_{\mathbf{yy}}$, respectively. Define the sum of these processes as $\mathbf{z}=\mathbf{x}+\mathbf{y}$, with covariance \begin{align} \mathbf{C}_{\mathbf{zz}}=\mathbf{C}_{\mathbf{xx}}+\mathbf{C}_{\mathbf{yy}}+\mathbf{C}_{\mathbf{xy}}+\mathbf{C}_{\mathbf{xy}}^\top. \end{align}
Assume that I know $\mathbf{C}_{\mathbf{xx}}$ and that I have observations of $\mathbf{z}$. Is there a procedure for inferring $\mathbf{C}_{\mathbf{yy}}$ and $\mathbf{C}_{\mathbf{xy}}$?