Let $C$ be a matrix; $v$ be a column vector; $P$, $\Delta$ are random matrices; $x$ is a random column vector.
$$Cvv^T - \mathbb{E}[P^Tx]v^T - \mathbb{E}[P^Tx v^T \Delta] + O(\Delta^2)= 0$$ $$C^TCv - C^T\mathbb{E}[P^Tx] - \mathbb{E}[\Delta C^TP^Tx] + O(\Delta^2)= 0$$
We also know that $\mathbb{E}[\Delta] = 0$ and $\Delta^T = \Delta$. I need to find a some expressions for C and v that satisfy both of the equations up to $O(\Delta^2)$. Do you have any idea?
