Three pairwise uncorrelated random variables

Question

Given $\xi$, $\eta$, $\zeta$ are pairwise uncorrelated, can we say, that $E(\xi\eta\zeta) = E\xi E\eta E\zeta$?

Answer 1

No. Counterexample:

Let $\xi$ and $\eta$ be iid with $\mathbb{P}\left(\xi=1\right)=\mathbb{P}\left(\xi=-1\right)=\frac{1}{2}$ and let $\zeta:=\xi\eta$.

Then $\xi$, $\eta$ and $\zeta$ are pairwise uncorrelated.

However $\mathbb{E}\xi\zeta\eta=\mathbb{E}\xi^2\eta^2=1\neq0=\mathbb{E}\xi\mathbb{E}\zeta\mathbb{E}\eta$.