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

Question

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.