Loading [MathJax]/extensions/TeX/boldsymbol.js

 

 

 

Definition of Correlation Functions and Standard Deviation

Let us analyze the problem by splitting up the correlation term into partial sums of the form \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 correlation term of the total variance can now be rewritten in terms of f_d \begin{equation*} \frac{2}{mn^2}\sum_{\alpha=1}^m\sum_{k < l}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle)= \frac{2}{n}\sum_{d=1}^{n-1} f_d \end{equation*}