Let $X_1$ ,$X_2$ be two r.v.'s with joint and marginal ch.f.'s $\phi_{X_1,X_2}$, $\phi_{X_1}$ and $\phi_{X_2}$.
By an example, show that
$\phi_{X_1,X_2} (t,t) = \phi_{X_1} (t) \phi_{X_2} (t)$ $\quad \forall t \in \Bbb R$
does not imply independence of $X_1$,$X_2$
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volkov
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Perhaps see http://math.stackexchange.com/questions/376511/a-criterion-for-independence-based-on-characteristic-function – user1576713 Sep 27 '14 at 13:44