statistics – Fisher information of joint distribution of transformed Normal distribution

Suppose $X_1=theta+epsilon_1$ and $$X_i=sqrt{gamma}X_{i-1}+sqrt{1-gamma}epsilon_i+theta(1-sqrt{gamma})$$
Where $gamma in (0,1)$ and $theta$ is the parameter of the model. Also $epsilon_1,epsilon_2,…epsilon_n$ are iid $N(0,1)$.

What is the Fisher information of this model and for what values of $gamma$ does it tensorise. I’ve tried using the Jacobian to find the joint distribution but I’m not sure, especially when determining for which values we have tensorisation. Any help would be much appreaciated.