Let be i.i.d. sampes from Let
Prove that and are uncorrelated for any
for simplicity let i=1....
Let's obtain the covariance between and
Now find the covariance between each pair, taking one term from each set.
ALL we need is for the for each , we don't need NORMALITY...
Next use the fact that all the variances are equal this becomes zero.
Changing 1 to i is easy, just sum over all the terms that's not i in the second sum.