For part 6.5a, the answer I have is that the joint distribution is also a Gaussian distribution. I understand that x(0) ... x(t+1) have Gaussian distributions because they are the linear transformations of other Gaussian random variables. However, I thought that Gaussian variables cannot always be assumed to be jointly gaussian.
Any guidance would be much appreciated.
Let $$\mathcal A = \begin{pmatrix} I \\ A \\ A^2 \\ \vdots \\ A^T \end{pmatrix}\qquad\text{and}\qquad z = \begin{pmatrix} x_0 \\ x_1 \\ \vdots \\ x_T \end{pmatrix}.$$ The goal is to find the distribution of $z$. Recursively plugging in yields $x_t = A^tx_0 + \sum_{\tau = 0}^tA^\tau w$. That is, in matrix notation, $$z = \mathcal Ax_0 + \mathcal C\mathcal Aw,$$ where $\mathcal C$ is the polynomial maker matrix. If we can assume that $x_0$ and $w$ are independent, we have that $z$ is $\mathcal N(\mathcal A\mu_0 \mathcal C\mathcal A(Q+\Sigma_0)\mathcal A'\mathcal C')$. In fact, 6.5a cannot be solved without imposing assumptions on the joint distribution of $x_0$ and $w$. We have to assume at least that the joint distribution exists; the distribution im question will be a normal distribution then