Let $X$ and $Y$ be two random variables. Let's denote $G_{X}$ and $G_{Y}$ their generating functions. We know that if $X$ and $Y$ are independent then $$ G_{X+Y}=G_{X}.G_{Y} $$
However, we know the reciprocal is not true. ($X$ and $Y$ following a Cauchy's law is a counter-example.)
However, is there any known example on random variable with discrete values ?