Sample Mean of AR(1) model Consider the AR(1) model with iid innovations with finite mean and variance. Also, let $X_0 = 0$.
\begin{align}
X_t = \phi X_{t-1} + \epsilon_t
\end{align}
The goal is to derive the asymptotic distribution of the sample mean, when $|\phi| < 1$ as well as when $\phi = 1$ or $-1$. From what I know, to derive such a result, one should know some theorems from a time series course. Yet, this is a practice question for my math stat class, so apparently there's a simple way of doing this that I'm missing. Any insight is appreciated.
Btw I already know the final answer, just not the steps. If it helps, the final answer for the first case is that $\sqrt{n}(\bar{X}-\frac{\mu}{1-\phi})$ is asymptotically normal with 0 mean and variance $\frac{\sigma^2}{(1-\phi)^2}$.
 A: FIRST STEP
Sometimes, patience and algebra are still required to obtain what we need to obtain. In your case, by repeated substitution as already suggested we get
$$X_t = \sum_{j=0}^t\phi^j\epsilon_{t-j}$$
and we note that, although not clearly stated in the question, here we have $E(\epsilon_t) = \mu$, not necessarily zero. The sample mean for a sample of size $T$ is therefore 
$$\bar X = \frac 1T\sum_{t=1}^TX_t = \frac 1T\sum_{t=1}^T\sum_{j=0}^t\phi^j\epsilon_{t-j}$$
Don't despair at this point. Patiently write out the internal sums for each $t=1,...T$ ($T$ is still finite) and you will see that you can re-arrange them as a sum in the innovations, each innovation being multiplied by a different constant term (although these constant terms will obviously form a recognizable pattern). So this will be a linear combination of i.i.d. random variables. So it will be a sum of independently but not identically distributed random variables...
SECOND STEP
So we have that
$$T\bar X = \sum_{t=1}^T\Big( \epsilon_t + \phi\epsilon_{t-1} + \phi^2\epsilon_{t-2}+...+\phi^{t-1}\epsilon_1\Big)$$
$$\begin{align} =& \epsilon_1 &\\
+&\phi\epsilon_1 +\epsilon_2 \\
+&...\\
+&\phi^{T-1}\epsilon_1+\phi^{T-2}\epsilon_2+...+ \epsilon_T\\
\end{align}$$
(reversing the order and summing per innovation)
$$=\epsilon_T + (1+\phi)\epsilon_{T-1} + (1+\phi+\phi^2)\epsilon_{T-2} +...+(1+\phi+\phi^2+...+\phi^{T-1})\epsilon_1$$
$$\Rightarrow \bar X = \frac 1T \sum_{t=1}^T\left[\left(\sum_{j=t}^T\phi^{T-j}\right)\epsilon_t\right] $$
So we see that the sample mean is a sum of independently but not identically distributed random variables. Therefore, we can invoke this variant of the classical Central Limit Theorem, and check if and when the Lindeberg and/or Lyapunov conditions hold (without the need to go into martingale theory, Gordin's conditions etc, which are the "time series course" material you mentioned that prove directly a CLT for dependent processes).
I would suggest to start with the case $\phi=1$.
