I want to prove the following:
If two RVs $X, Y$ are sub-independent, i.e., $\phi_{X+Y}(t) = \phi_X(t)\phi_Y(t), t\in\mathbb{R}$ then $X, Y$are uncorrelated. Keep $Cov(X,Y) = E(XY)-E(X)E(Y) = 0$ in mind, I'm considering the way to connect $E(X)$ with characteristic function $\phi_X(t)$. In fact, $\phi_{X}'(0) = iE(X), \phi_{Y}'(0) = iE(Y).$ Finding $E(XY)$ will end the proof, but I could not proceed further.
Any idea or tips would be really helpful.
The trick is to introduce auxiliary variables $X'$ and $Y'$ with known joint distribution. Define $X'$ and $Y'$ to be independent but with the same marginal distributions as $X$ and $Y$ respectively. Then $$ \phi_{X'+Y'}(t)\stackrel{(a)}=\phi_{X'}(t)\phi_{Y'}(t) \stackrel{(b)}=\phi_X(t)\phi_Y(t)\stackrel{(c)}=\phi_{X+Y}(t); $$ in (a) we use independence of $X'$ and $Y'$, in (b) we use equality of marginal distributions; and (c) is the definition of sub-independence. Therefore, by uniqueness of characteristic functions, $X'+Y'$ has the same distribution as $X+Y$. In particular, $$ E(X'+Y')^2=E(X+Y)^2 $$ (assuming second moments exist), which in turn implies $$E(XY)=E(X'Y')\stackrel{(a)}=E(X')E(Y')\stackrel{(b)}=E(X)E(Y).$$