limiting distribution of Xbar considering an autoregressive process

Question

Here is the question:

Consider the stationary Gaussian autoregressive process of order 1, $X_{i+1} − μ = ρ(X_i − μ) + \sigma Z_i$, where $Z_i$ are iid N(0, 1). Find the limiting distribution of $\overline{X}$ assuming |ρ| < 1.

Here is what I have so far:

$X_{i+1} − μ = ρ(X_i − μ) + \sigma Z_i$

$\sum (X_{i+1} − μ) = \sum (ρ(X_i − μ) + \sigma Z_i)$

$\sum(X_{i+1}) − nμ = ρ\sum(X_i) − n\rhoμ+ \sigma \sum(Z_i)$

$(X_1-X_1+X_2+X_3+...+X_{i+1} − nμ =ρ\sum(X_i) − n\rhoμ+ \sigma \sum(Z_i)$

by expanding sum

$\sum X_i+X_{i+1}-X_1-n\mu=ρ\sum(X_i) − n\rhoμ+ \sigma \sum(Z_i)$

dividing both sides by n

$\overline{X}+\frac{X_{i+1}-X_1}{n}-\mu=ρ\overline{X} − \rhoμ+ \sigma \overline{Z}$

$\overline{X}-\rho\overline{X}= \mu -\rho \mu +\sigma \overline{Z}-\frac{X_{i+1}-X_1}{n}$

$\overline{X}(1-\rho)=\mu(1-\rho)+\sigma \overline{Z}-\frac{X_{i+1}-X_1}{n}$

$\overline{X}= \mu+\frac{\sigma \overline{Z}-\frac{X_{i+1}-X_1}{n}}{1-\rho}$

But now I am stuck... I know that $\overline{Z}$ converges to N(0,1) but I am not sure what to do now.

Any help is appreciated!

Answer 1

Correction: sorry, my previous explanation was wrong.

Something is not right in your computation as the final term converges to zero and not to a random variable. I guess you want to consider $\sqrt{n}(\bar{X}-\mu)$ to get a CLT. Again, using Slutskys Theorem should work, but please clarify your intention first.