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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!