is the concatenation of White noise process samples in a vector white also?

Question

Given that $X_t \sim N\left( {0,{I_2}{\sigma ^2}} \right)$ is a bivariate white noise process, what are the characteristic of the new process ${Y_t} = \left[ {\begin{array}{*{20}{c}}{{X_t}}\\ \vdots \\{{X_{t + n - 1}}}\end{array}} \right]$ , is it white also , what is its covariance matrix?

Answer 1

A bit of intuition of why $Y_t$ is not a white process. If it were white it would mean that the two variables $Y_{t+\tau}$ and $Y_{t}$ are uncovariated for any $\tau\neq 0$. But for example

$$ Y_{t+1}= \begin{bmatrix} X_{t+1} \\ \vdots\\ X_{t+n} \end{bmatrix} $$

is clearly correlated with $Y_t$ (a lot of the elements are actually the same). So clearly it is not white. Speaking of covariance matrix, do you refer to the $2\times n$ vectorized version?