Cramér theorem for Gamma random variables

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In this paper we discuss the following problem: given a random variable Z=X+Y with Gamma law such that X and Y are independent, we want to understand if then X and Y each follow a Gamma law. This is related to Cramér's theorem which states that if X and Y are independent then Z=X+Y follows a Gaussian law if and only if X and Y follow a Gaussian law. We prove that Cramér's theorem is true in the Gamma context for random variables leaving in a Wiener chaos of fixed order but the result is not true in general. We also give an asymptotic variant of our result.