Is it possible to answer this problem with standard Central Limit Theorem or should we use Lindeberg-Feller CLT?

Question

I have the following problem on my Statistics I problem set:

Suppose that $X_t = \mu + U_t$, where $U_t = V_t + \rho V_{t-1}$ and $V_t$ are iid standard normal variables.

1. Apply a CLT to find the limiting distribution of $\sqrt{n} (\bar{X}_n -\mu)$
2. Let $\hat{\theta}_n = (\hat\mu_n, \hat\rho_n)$ be the MLE for $\theta = (\mu, \rho)$. Find the asymptotic distribution of $\sqrt{n}(\hat{\theta}_n - \theta)$.
3. Compare the asymptotic distributions of $\sqrt{n}(\hat\mu_n - \mu)$ and $\sqrt{n}(\bar{X}_n - \mu)$. Explain your answer.

I cannot prove part 1, since $X_{t}$ variables are not iid, as required by the standard CLT. I understand that we should use Lindeberg-Feller CLT or something stronger to prove this result.

Can anyone do it with standard CLT?

Answer 1

Write $$ \sum_{t=1}^nU_t=\sum_{t=1}^nV_t+\rho\sum_{t=1}^nV_{t-1}. $$ In the second sum, do the change of index $j\leftarrow t-1$ to get $$ \sum_{t=1}^nU_t=\sum_{j=1}^nV_j+\rho\sum_{j=0}^{n-1}V_{j}=\rho V_0+V_n+\left(1+\rho\right)\sum_{j=1}^{n-1}V_{j}. $$ It follows that $$ \sqrt n\left(\overline{X_n}-\mu\right)=\frac 1{\sqrt n}\sum_{t=1}^nU_t =\frac{1+\rho}{\sqrt n}\sum_{j=1}^{n-1}V_{j}+\frac{1}{\sqrt n}\left(\rho V_0+V_n\right). $$ The random variable in the right hand side is Gaussian and centered. It remains to compute the variance and itslimit.