I would appreciate some help with showing:
Cov(S[i= 1..n] Xi, S[j= 1..m] Yj) = S[i= 1.. n]S[j= 1..m](Cov(Xi, Yj)
Where S[i= 1..n] denotes summation terms i from 1 to n; likewise for S[j= 1..m].
= Cov(X1 + X2 + ... + Xn, Y1 + Y2 + ... + Yn)
Now there might be a correlation with say X2 and Y5 so:
= Cov(X1, Y1) + Cov(X1, Y2) + ... + Cov(X1, Ym) +
Cov(X2, Y1) + Cov(X2, Y2) + ... + Cov(X2, Ym) +
Cov(Xn, Y1) + Cov(Xn, Y2) + ... + Cov(Xn, Ym)
Applying the definition of Cov(X, Y) to the above I obtain the required result.
However I do not really understand why I need to sum the covariances for all the pairs as listed above. I know that if there is no correlation between an X, Y pair then the covariance will be 0 so they are not contributing to the sum. But why is the covariance of the sums equal to the sum of the covairances.
A proof would be great as it would help me understand what is really going on.