# Derivation of the Variance of Autocorrelation

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## Main Question or Discussion Point

Hi everyone in this link (https://stats.stackexchange.com/questions/226334/ljung-box-finite-sample-adjustments) I see the variance of autocorrelation related to specific lag is demonstrated in the following: $$Var(r_k) = \frac {\sum_{t=k+1}^n a_t*a_{t-k}} {\sum_{t=1}^n a_t^2}$$ where ##r_k## is autocorrelation at relevant lag, ##n## is the number of data set and ##a_t## is error. Could help me prove the formula I mentioned above?

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Hi everyone in this link (https://stats.stackexchange.com/questions/226334/ljung-box-finite-sample-adjustments) I see the variance of autocorrelation related to specific lag is demonstrated in the following: $$Var(r_k) = \frac {\sum_{t=k+1}^n a_t*a_{t-k}} {\sum_{t=1}^n a_t^2}$$ where ##r_k## is autocorrelation at relevant lag, ##n## is the number of data set and ##a_t## is error. Could help me prove the formula I mentioned above?
I have seen the following formula whereas $$r_k = \frac {\sum_{t=k+1}^n a_t*a_{t-k}} {\sum_{t=1}^n a_t^2}$$ $$Var(r_k) = \frac {n-k}{n*(n+2)}$$ where $r_k$ is the autocorrelation at relevant lag, $n$ is the number of points in the data set, and $a_t$ is the error.