How to describe correlated noise after the signal is oversampled?
Assume that the input to a simple up-sampler (say by $L$) is $x[n]$, hence 
$$y[n] = \begin{cases} { x[n/L] ,\qquad n=kL \\ 0 , \qquad \qquad \ \text{  else } }\end{cases}$$
If you compute the auto-correlation sequence, defined as 
$$\phi_{xx}(i,j) = \mathbb{E}(x[i]x[i+j])$$
for an iid white Gaussian process, we have that 
\begin{align}
\phi_{xx}(i,j) &amp;= \begin{cases} \sigma^2, \qquad &amp; \text{if } j=0 \\
0, \qquad &amp; \text{else } \end{cases} \\
\phi_{yy}(i,j) &amp;= \begin{cases} \sigma^2, \qquad &amp; \text{if } j=0 \text{ and }i = kL \\
0, \qquad &amp; \text{else } \end{cases}
\end{align}
Unlike i.i.d Gaussian, where $\phi_{xx}(i,j)$ is determined by the lag $j-i$; $\phi_{yy}(i,j)$ is not fully determined by the lag. That said, $y[n]$ is not WSS, whereas $x[n] $ is. With respect to correlation, the quantities $\mathbb{E}(x[n]x[m]) = 0$ for $n \neq m$ is due to the uncorrelation assumption. You can say the same thing on $y[n]$, namely $\mathbb{E}(y[n]y[m]) = 0$ for $n \neq m$.