There's this proposition I encountered in a paper on Bayesian data analysis. I am trying to use it to ease some computations for a mixed effects model. It goes as follows:
$\textbf{Proposition 1.}$ Let $z_d, d = 1, \ldots, D - 1,$ be independent Gaussian random variables with different means $\mu_d$ and variances $\sigma^2_d$, and let $u \sim \mathcal{N}(0, \gamma)$. Then, the multivariate random variable $\mathbf{y}$, defined as:
\begin{align} y_1 &= z_1 + u \\ y_2 &= z_2 + u \\ &\vdots \\ y_{D-1} &= z_{D-1} + u \end{align}
follows a multivariate Gaussian with mean vector $\boldsymbol{\mu}$ and covariance matrix $\Sigma$ whose elements are:
\begin{equation} \Sigma_{dd} = \sigma^2_d + \gamma, \quad d = 1, \ldots, D - 1 \end{equation} \begin{equation} \Sigma_{dj} = \gamma, \quad d \neq j \end{equation}
Does this proposition still hold if we replace the random effect, $u$, with a spatial random effect like a conditional autoregressive term? In particular, I want to include full conditional distributions that are given by:
\begin{equation} s_i | \boldsymbol{s}_{-i}, \tau_s \sim \mathcal{N}\left(\sum_{i \in \delta_i} \frac{s_i}{|\delta_i|}, \frac{1}{\tau_s |\delta_i|}\right), \end{equation}
where $|\delta_i|$ is the number of neighbors defined through an adjacency matrix, $\tau_s$ is an effect specific precision term, and the $-i$ subscript denotes all elements in the vector $\boldsymbol{s}$ that are not the $i^{th}$. In this case, the mean and variance are not equal for each $y_d$. My question is if we can conclude the following nonetheless:
\begin{equation} \boldsymbol{\mu} = (\sum_{i \in \delta_1} \frac{s_i}{|\delta_1|}, \cdots, \sum_{i \in \delta_{D-1}} \frac{s_i}{|\delta_{D-1}|})^T \end{equation} \begin{equation} \Sigma_{dd} = \sigma^2_d + \frac{1}{\tau_s |\delta_i|}, \quad d = 1, \ldots, D - 1 \end{equation} \begin{equation} \Sigma_{dj} = \frac{1}{\tau_s |\delta_i|}, \quad d \neq j \end{equation}
(Also, what would be a good starting place to learn about this type of mathematics? I need to use it often in my studies.)