If we have the process $\displaystyle X_t=e^{\theta t}x+\int_0^t e^{ \theta (t-s)}dB_s$, where x is a constant. How do I show that if $\displaystyle \theta<0$ then, when $\displaystyle t\to\infty$, $\displaystyle X_t$ converges in distribution toward a Gaussian distribution?

Can I use the fact that $\displaystyle E(X_t)=xe^{\theta t}$ which goes to 0 when $\displaystyle t\to\infty ,\theta<0$? This implies that $\displaystyle X_t$ converges to a Gaussian process?