Suppose I have a random gaussian vector $\xi=[\xi_{1}(t),\xi_{2}(t),\xi_{3}(t)]^{T}$ with zero mean $\left(\textbf{E}\left(\xi_{1}(t)\right)=\textbf{E}\left(\xi_{2}(t)\right)=\textbf{E}\left(\xi_{3}(t)\right)=0\right)$ and that the covariances of each $\xi$ are delta-correlated. Namely
$$ \textbf{E}(\xi_{i}(t)\xi_{j}^{*}(t^{\prime})) = \delta_{ij}\delta(t-t^{\prime}) $$
where the elements of $\xi$ can be complex numbers. If I define $Y$ such that $Y_{j}=\xi_{j}-\xi_{j-1}$, how can I calculate the covariance of $Y$? Is it also true that
$$ \textbf{E}(Y_{i}(t)Y_{j}^{*}(t^{\prime})) = \delta_{ij}\delta(t-t^{\prime}) $$
?
The covariance of $Y_i$ and $Y_j$ is
$$\begin{align} \text{Cov}\left(Y_i(t),Y_j(t')\right)&=\text{Cov}\left(\xi_i(t)-\xi_{i-1}(t),\xi_j(t')-\xi_{j-1}(t')\right)\\\\ &=\text{Cov}\left(\xi_i(t),\xi_j(t')\right)-\text{Cov}\left(\xi_{i-1}(t),\xi_j(t')\right)\\\\ &-\text{Cov}\left(\xi_i(t),\xi_{j-1}(t')\right)+\text{Cov}\left(\xi_{j-1}(t),\xi_{j-1}(t')\right)\\\\ &=\left(\delta_{i,j}-\delta_{i-1,j}-\delta_{i,j-1}+\delta_{i-1,j-1}\right)\delta(t-t')\\\\ &=(2\delta_{i,j}-\delta_{i,j-1}-\delta_{i-1,j})\delta(t-t')\\\\ \end{align}$$