Since our random numbers, which are typically generated via a linear congruential algorithm, are never fully independent, we can then define an important test which measures the degree of correlation, namely the so-called auto-correlation function defined previously, see again Eq. (11). We rewrite it here as $$ \begin{equation*} C_k=\frac{f_d} {\sigma^2}, \end{equation*} $$ with \( C_0=1 \). Recall that \( \sigma^2=\langle x_i^2\rangle-\langle x_i\rangle^2 \) and that $$ \begin{equation*} f_d = \frac{1}{nm}\sum_{\alpha=1}^m\sum_{k=1}^{n-d}(x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,k+d}-\langle X_m \rangle), \end{equation*} $$

The non-vanishing of \( C_k \) for \( k\ne 0 \) means that the random numbers are not independent. The independence of the random numbers is crucial in the evaluation of other expectation values. If they are not independent, our assumption for approximating \( \sigma_N \) in Eq. (3) is no longer valid.