<s>[INST] Prove variance of mean estimator of an AR(1) model I need to solve the proof as given in the screenshot below. I had tried to do something (under My solution) but I don't know how to proceed with the proof. I'm not sure whether I did something wrong or mistake. Please help me to continue with the proof. Any help will be greatly appreciated! Thanks!

 [/INST] Compute directly (not the best homework question, if that's what this is):
\begin{align*}
Var(\bar{Y}) &= Var(\frac{1}{n} \sum_{t=1}^n Y_t) \\
             &= \frac{1}{n^2} Var( \sum_{t=1}^n Y_t) \\
             &= \frac{1}{n} \gamma(0) + 2 \frac{1}{n} \sum_{h = 1}^{n-h} \frac{n-h}{n} \gamma(h),
\end{align*}
where $\gamma(h) = \frac{\beta^h}{1-\beta^2}$ is the autocovariance function at lag $h$.
Substituting $\gamma(0) = \frac{1}{1-\beta^2}$ gives the first term in your sum
$$ 
\frac{1}{n} \gamma(0) = \frac{1}{n} \frac{1}{1-\beta^2}.
$$
Similarly,
\begin{align*}
2 \frac{1}{n} \sum_{h = 1}^{n-h} \frac{n-h}{n} \gamma(h) &= \frac{2}{n^2(1-\beta^2)}  \sum_{h=1}^{n-h} (n-h) \beta^h \\
&=  \frac{2}{n^2(1-\beta^2)}  \sum_{h=1}^{n-h} \sum_{j=1}^{n-h} \beta^j \\ 
&=  \frac{2}{n^2(1-\beta^2)}  \sum_{h=1}^{n-h} ( \frac{1-\beta^{n+1-h}}{1-\beta}) \\
&=  \frac{2}{n^2(1-\beta^2)(1-\beta)}  \sum_{h=1}^{n-h} ( 1-\beta^{n+1-h} )\\
&= \cdots.
\end{align*}
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