Variance of a sum of correlated variables.

Question

http://www.maths.qmul.ac.uk/~ob/MTH6121/moddocs/chap1_4.pdf

I'm currently reading over some notes, linked above, and I've come across a difficulty with proposition 1.27(page 12).

Underneath, it says the proof follows from 1.24. I can very easily prove 1.24, but for the life of me cannot get through this one. I just can't fully understand or appreciate where the $X_j$ counter comes from.

I hope that made sense,

Thanks.

Answer 1

You have $$ Var\left(\sum_{i}X_i\right) = Cov\left(\sum_{i}X_i,\sum_{j}X_j \right)$$ and you use different indices in order to get $X_1 X_2$, $X_1 X_3$ etc. You can't achieve a double sum by using only one index.

EDIT: It's like dealing with $\left(\sum_j X_j\right)^2$ where you use $\left(\sum_j X_j\right)^2 = \left(\sum_j X_j\right)\left(\sum_i X_i\right) = \sum_j\sum_i X_iX_j$ in order to make use of eventual independencies in proofs.