 Solution

Since $X_1$ is the sum of two independent centered Gaussian innovations,For any nonzero $a\in {\mathbb{R}}^n$,Expanding each $X_t={\epsilon }_t+\theta {\epsilon }_{t-1}$ expresses this as ${\sigma }^2$ times a sum of squared innovation coefficients. If all coefficients vanished, the coefficient of the latest innovation gives $a_n=0$, and backward induction gives every $a_t=0$, a contradiction. Hence $a^T\Sigma a>0$ and the covariance matrix is positive definite.