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.