# Is a Stationary VAR Process with Zero Mean Gaussian Innovations a Gaussian Stationary Process?

Consider the stationary VAR process

$${\bf X}_t = \sum_{\tau = 1}^{L} A_\tau {\bf X}_{t-\tau} +{\bf \epsilon}_t$$

If the innovations $\epsilon_t \sim MVN({\bf 0},\Sigma)$ then is ${\bf X}_t$ a Gaussian stationary process?

Is it correct that due to the invertibility of the VAR into an MA and observing that ${\bf X}_t$ is the sum of zero mean MVN random variables the above is true, or is there a flaw somewhere in this.

• Duplicate of question already asked on dsp.SE – Dilip Sarwate Nov 29 '13 at 19:40
• Yes, someone there told me to post it here – rwolst Nov 29 '13 at 21:28

It depends what distribution you start the process in. Yes if the initial distribution is $$\left[\begin{array}{c} \mathbf{X}_1 \\ \vdots \\ \mathbf{X}_L \end{array}\right] \sim \text{Normal}\left( \mathbf{0}, \left[\begin{array}{cccc} \Gamma(0) & \Gamma(-1) & \cdots & \Gamma(1-L) \\ \Gamma(1) & \ddots & \cdots & \Gamma(2-L) \\ \vdots & \vdots & \cdots & \vdots \\ \Gamma(L-1) & \Gamma(L-2) & \cdots & \Gamma(0) \end{array}\right] \right).$$
First, causality (not invertibility) lets you write the model as a sum of the infinite past of $\epsilon$ terms. The above condition doesn't posit the existence of an infinitely long history of data, but if you wanted to go that way, then yes, that will work, too. It is because linear combinations of Normal vectors end up being Normal as well.