We introduce now a variable \( d=\vert j-k\vert \) and rewrite
$$ \frac{2}{mn^2}\sum_{i=1}^{m} \sum_{j < k}^{n}\tilde{x}_{ij}\tilde{x}_{ik}, $$in terms of a function
$$ f_d=\frac{2}{mn}\sum_{i=1}^{m} \sum_{k=1}^{n-d}\tilde{x}_{ik}\tilde{x}_{i(k+d)}. $$We note that for \( d=0 \) we have
$$ f_0=\frac{2}{mn}\sum_{i=1}^{m} \sum_{k=1}^{n}\tilde{x}_{ik}\tilde{x}_{i(k)}=\sigma^2! $$