 Solution

Let $\ell (\theta ;x)=\log p(x\mid \theta )$. The maximum-likelihood estimator maximizes this function; the score function is $U={\partial }_{\theta }\ell$, the observed Fisher information is $\hat{j}=-{\partial }_{\theta }^2\ell (\hat{\theta };x)$, and the Fisher information is $I_n(\theta )={\mathbb{E}}_{\theta }{U}^2=-{\mathbb{E}}_{\theta }{\ell }^{\prime \prime }$ under the stated differentiation regularity. Differentiating normalization gives $\mathbb{E}U=0$. For iid observations the score is a sum of iid centered terms with variance $I_1$, so the central limit theorem gives $U/\sqrt{n{I}_1}\Rightarrow N(0,1)$ when $0<I_1<\infty$. Under the additional usual consistency, identifiability and interior-MLE regularity, $\sqrt{I_n}(\hat{\theta }-\theta )\Rightarrow N(0,1)$. Differentiation under the integral alone does not imply all these MLE conditions.

For the specified autoregression, index the innovations so $X_i=\theta X_{i-1}+E_i$, with $x_0=1$. Conditional normal densities giveSet $A=\sum x_{i-1}^2\ge 1$, $B=\sum x_{i-1}{x}_i$ and $C=\sum x_i^2$. Completing the square givesThe likelihood ratio between two data vectors is independent of $\theta$ precisely when their coefficients $A,B$ agree, because its log is a constant plus $\theta \Delta B-{\theta }^2\Delta A/2$. The positive-density likelihood-ratio criterion therefore makes $(A,B)$ minimal sufficient. Since $(A,B)\leftrightarrow (\hat{j},\hat{\theta })$ is bijective, the requested pair is also minimal sufficient.

With the flat improper prior, the posterior kernel is $\exp [-A(\theta -\hat{\theta }{)}^2/2]$, which is integrable because $A\ge 1$. ThusConditionally on the observations, $S(\theta -\hat{\theta })$ is exactly standard normal, as is $E_1$. This is a posterior statement, not a claim that the estimator standardized by the random information is an exact normal sampling pivot.