I meet a problem saying "random variables $X,Y$ are independent iff the characteristic function of $X+Y$ is equal to the product of characteristic functions of $X$ and $Y$." It is not hard to the the forward direction is true. I wonder how could we show the backward?
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What is your source for the problem asking to prove that "Random variables X, Y are independent iff the characteristic function of X+Y is equal to the product of characteristic functions of X and Y"? (Note that the assertion: "If random variables X, Y are independent iff the characteristic function of X+Y is equal to the product of characteristic functions of X and Y." is absurd since one cannot write "If P iff Q".) – Did Dec 12 '16 at 14:35
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You cannot as it is not necessarily true. For a counter example, let $X$ be a Cauchy random variable and let $X = Y$ a.s. The characteristic function of the Cauchy with location $x_0$ and scale $\gamma$ is given by
$$ \phi_X(t) = e^{i x_0 t + \gamma \lvert t\rvert} $$ and thus $$ \phi_X(t)\phi_Y(t) = \left(e^{i x_0 t + \gamma \lvert t\rvert}\right)^2 = e^{ix_0\cdot 2t + \gamma \lvert 2t\rvert } = E[e^{it\cdot 2X}] = E[e^{it(X+Y)}] = \phi_{X+Y}(t), $$ but $X$ and $Y$ are not independent. (Example motivated by user @Did)
