$\{\mathbf{Z}_{\mathbf{i}}\}_{\mathbf{i} \in Z^{d}}$ is a strictly stationary process, following an isotropic short memory dependence with the covariance matrix given by $\big\{ C_{k,l}(\lVert \mathbf{i}-\mathbf{j} \lVert) \big\}_{k,l=1}^{p+q} $ where $ \forall \mathbf{i}\neq \mathbf{j}$, $$C_{k,l=1}(\lVert \mathbf{i}-\mathbf{j} \lVert)=\operatorname{Cov}(<\mathbf{Z_{i}},e_{k}>, <\mathbf{Z_{j}},e_{l}>) =\sigma_{k}\sigma_{l}\exp{(-an\lVert \mathbf{i}-\mathbf{j} \lVert)} $$.
I assume that : $\forall \mathbf{i_{1}},\mathbf{i_{2}}\neq\mathbf{j_{1}},\mathbf{j_{2}} , \forall k,l,t,u \in \{1,2,...,p+q\}$ $\operatorname{Cov}(Z_{\mathbf{i_{1}}}^{(k)}Z_{\mathbf{i_{2}}}^{(u)}, Z_{\mathbf{j_{1}}}^{(l)}Z_{\mathbf{j_{2}}}^{(t)})=\operatorname{Cov}(Z_{\mathbf{i_{1}}}^{(k)},Z_{\mathbf{j_{1}}}^{(l)})\operatorname{Cov}(Z_{\mathbf{i_{2}}}^{(u)},Z_{\mathbf{j_{2}}}^{(t)}) + \operatorname{Cov}(Z_{\mathbf{i_{1}}}^{(k)},Z_{\mathbf{j_{2}}}^{(t)})\operatorname{Cov}(Z_{\mathbf{i_{2}}}^{(u)},Z_{\mathbf{j_{1}}}^{(l)}).$
I have two questions:
1 - is it necessary to set this condition to control the convergence of
$n^{-d}\sum\limits_{l,t=1}^{p+q}\sum\limits_{\parallel\mathbf{i-j}\parallel > 0}u^{(l)}u^{(t)}\operatorname{Cov}(\textbf{Z}_{\textbf{i}}^{(k)}\textbf{Z}_{\textbf{i}}^{(l)}, \textbf{Z}_{\textbf{j}}^{(k)}\textbf{Z}_{\textbf{j}}^{(t)})$, if yes I would like to have an article that can clarify me.
2- I have always seen this assumption of covariance decomposition on error processes [Fransco and fernandez(2007) ] and I don't know if it is plausible for all processes. if yes I would like to have an example in an article or an example from you.
Thank you for your reply.