Let us analyze the problem by splitting up the correlation term into partial sums of the form:

$$f_d = \frac{1}{n-d}\sum_{k=1}^{n-d}(x_k - \bar x_n)(x_{k+d} - \bar x_n)$$

The correlation term of the error can now be rewritten in terms of $f_d$

$$\frac{2}{n}\sum_{k < l} (x_k - \bar x_n)(x_l - \bar x_n) = 2\sum_{d=1}^{n-1} f_d$$

The value of $f_d$ reflects the correlation between measurements separated by the distance $d$ in the sample samples. Notice that for $d=0$, $f$ is just the sample variance, $\mathrm{var}(x)$. If we divide $f_d$ by $\mathrm{var}(x)$, we arrive at the so called autocorrelation function

$$\kappa_d = \frac{f_d}{\mathrm{var}(x)}$$

which gives us a useful measure of pairwise correlations starting always at $1$ for $d=0$.