Hey, can someone could help me with this proof?
If and X,Y are independent random variables, then
Yes, I saw once a proof similar to that. Not quite sure I can get it formally together, but remember independence implies that the covariance is zero. This further, in general, implies mean independence, that is,
Now, without going into probability limits, when you just multiply the series by 1/N and take the normal limit, you have by the law of large numbers .
For independence, as shown, it follows that or again expressed in limits