Derivation of the Variance of Autocorrelation

[mertcan]

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?

 

[mertcan]

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?

Guys sorry for wrong question. Please let me rectify it.
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.

I have searched the internet for the proof for variance equation, but I haven't found it. Could anyone help me prove the formula I mentioned above?