Rewriting the covariance term

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

$$f_0=\frac{2}{mn}\sum_{i=1}^{m} \sum_{k=1}^{n}\tilde{x}_{ik}\tilde{x}_{i(k)}=\sigma^2!$$

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