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This question is a natural continuation of the following question: How do I construct the score process of a Markov model and verify that it is a Martingale? In this problem, we set up as follows:

Consider the following autoregressive model: $$ X_{t+1} = \alpha_0 + \beta_0 (X_t - \alpha_0) + W_{t+1}, $$ where $-1 < \beta_0 < 1$ and $W_{t+1}$ is distributed as a normal with mean zero and variance one.

The question constructs a log-likelihood process and the associated score process:

$$ s_t(\theta \mid \textbf X) = \begin{bmatrix} (1 - \beta_0) \sum_{j=1}^t W_j \\ \sum_{j=1}^t W_j (X_{j-1} - \alpha_0) \end{bmatrix}. $$

After verifying that the score process is a Martingale, a natural follow is to compute the covariance matrix for a central limit approximation for this process and find the inverse of the matrix. How would I do this? Also, how would I interpret the one-one component of the inverse?

After looking at the Wikipedia article on the Martingale central limit theorem, it seems like this may be a little difficult given that the statement of the theorem requires the difference sequence to be uniformly bounded. Am I missing something or is there a generalization that I'm missing?

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