I am trying to fully understand the derivation of the covariance of the innovation vector, however I am stucked conceptually at a point. I will show you my reasoning and where I am stuck (if someone find something wrong I kindly ask for corrections):
Let's write the true measurement model and the measurement model after the state time prediction from epoch $k-1$ to $k$:
$z_k = H_k x_k + \epsilon_k~~~~~~~~~~\epsilon_k\sim N(0,R_k)$
and
$z_k = H_k \hat{x}^-_k + \upsilon_k$ where
$\upsilon $ accounts for the $measurement$ and $prediction$ noise such that $\sim N(0,S_k)$
Therefore the KF innovation vector (assumed column) is the following:
$\gamma = z_k-H_k \hat{x}^-_k~~~~~~~~~~\gamma_k\sim N(0,S_k)$
In this sense, the covariance of $\gamma$ is expressed as:
$S_k= E[(\gamma - \underbrace{E[\gamma]}_0)(\gamma - \underbrace{E[\gamma]}_0)^T ]$
$S_k = E[\gamma\gamma^T] = E[(z_k-H_k \hat{x}^-_k)(z_k-H_k \hat{x}^-_k)^T]$
$S_k = E[z_kz_k^T-\color{blue}{z_k(H_k \hat{x}^-_k)^T}-\color{blue}{H_k \hat{x}^-_kz_k^T} +H_k {\hat{x}^-_k}{\hat{x}^-_k}^TH^{T}_k]$
Now here I don't really know how to get rid of the blue terms (if is possible). Because if I do, then I will have:
$S_k = E[z_kz_k^T] + H_kE[{\hat{x}^-_k}{\hat{x}^-_k}^T]H^{T}_k$
$S_k = H_kP^{-}_kH^{T}_k + R_k$ which is the final expression (rearranged)
Any opinion will be highly appreciated.