Introducing the correlation function

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.

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