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?