## convergence in distribution

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

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