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 \( d=0 \).