can someone please explain what the covariance of the following is?
$$ \text{Cov}\big(\sum_{i}^{m}{X_{ik}},\sum_{k}^{o}\sum_{i}^{m}{X_{ik}}\big) $$
In Wolfram it says
$$ \text{Cov}\big(\sum_{i}^{m}{X_{i}},\sum_{j}^{n}{Y_{j}}\big) = \sum_{i}^{m}\sum_{j}^{n}{}\text{Cov}\big(X_{i},Y_{j}\big) $$
but each RV is being summed once. What if one is being summed twice?
Thanks
$$ \begin{align} \text{Cov}\Big(\sum \limits_i^m X_{ik} , \sum \limits_v^o \sum \limits_t^m X_{tv} \Big) & = \text{Cov}\Big(\sum \limits_i^m X_{ik} , \sum \limits_v^o \sum \limits_t^m X_{tv} \Big) \\ & = \text{Cov}\Big(\sum \limits_i^m X_{ik} , \sum \limits_v^o Y_{mv} \Big) \\ & = \sum \limits_i^m \sum \limits_v^o \text{Cov}\big(X_{ik},Y_{tm}\big)\\ & = \sum \limits_i^m \sum \limits_v^o \text{Cov}\big(X_{ik},\sum \limits_t^m X_{tv} \big)\\ & = \sum \limits_i^m \sum \limits_v^o \sum \limits_t^m\text{Cov}\big(X_{ik}, X_{tv} \big)\\ \end{align} $$
[1] Using the hint from @Max.
[2] Use the left-hand side of identity (18) with the right-hand side of identity (21) from Wolfram.
[3] Replace substitute variable with original value.
[4] Use the right-hand sides of identity (18) and (21) from Wolfram.