Let $\xi,\eta$ two random variables.
How can I prove that $\xi,\eta$ are independent if the following condition holds:
- there exists $C>0$ such that $|\xi|\le C$ and $|\eta| \le C$
- $\mathbf{E}(\xi^n\eta^m) = \mathbf{E}(\xi^n)\mathbf{E}(\eta^m), \forall n,m \in \mathbb{N}$.
What counterexamples show that the result does not hold if we remove either assumption 1. or assumption 2.?
Heh. My first reaction was this:
The result is simply not true. Let $\xi=1$, $\eta=1$.
But that's wrong - curiously those two variables are independent, even though $\xi=\eta$. (If you're saying to yourself "how can they be independent when they're equal?" you should look at the definition of independence...)
A counterexample without (2) is easy: Say $P(\xi=1)=P(\xi=0)=1/2$ and let $\eta=\xi$. (The wrong counterexample above shows you can't just say they're not independent because they're equal; verify that the definition does not hold!)
A counterexample satisfying (2) but not (1) seems less trivial.
To prove the theorem, applying the Weierstrass approximation theorem on $[-C,C]$ shows that $E(f(\xi)g(\eta))=F(f(\xi))E(g(\eta))$ for any continuous functions $f$ and $g$. This implies that $\xi$ and $\eta$ are independent. How it implies that depends on what you know - if you've learned basic properties of the "characteristic function" it follows easily, if not approximate $\Bbb 1_E$ and $\Bbb 1_F$ almost everywhere by uniformly bounded continuous functions and apply dominated convegence or something similar.