We introduce then a correlation function \( \kappa_d=f_d/\sigma^2 \). Note that \( \kappa_0 =1 \). We rewrite the variance \( \sigma_m^2 \) as
$$ \begin{align*} \sigma^2_{m}& = \frac{\sigma^2}{m}\left[1+2\sum_{d=1}^{n-1} \kappa_d\right]. \end{align*} $$The code here shows the evolution of \( \kappa_d \) as a function of \( d \) for a series of random numbers. We see that the function \( \kappa_d \) approaches \( 0 \) as \( d\rightarrow \infty \).
In this case, our data are given by random numbers generated for the uniform distribution with \( x\in [0,1] \). Even with two random numbers being far away, we note that the correlation function is not zero.