Thread: law of large number

I have problem with the following question

Let be independent and identically distributed random variables with . Show that


where

I proceed this way:
I want to know wether this is true or not.

???

Now since are independent and identically distributed, then




so finally

I'll use X's instead



So

Thus

But each of these are the same, producing

Thanks for your reply,

I want to know what about this, is this true ?



Thanks

Both are S_n, as long as you condition on S_n, you know S_n, the rest of that sequence is unnecessary.

E(X|X)=X=E(X|X,Y)

Same problem : http://www.mathhelpforum.com/math-he...-v-136142.html

And I don't agree with you going to this step :



equality of the expectations doesn't mean equality of the conditional expectations.

As for proving that , remember that if we have two sigma-algebras (or sigma-algebras generated by random variables) such that , then for a rv X,

now consider and and you have your equality.

(why is there the inclusion ? because if is a measurable function of and is a measurable function of )

If you don't know what a sigma-algebra is, then just forget my post