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.