Cram\'er theorem for Gamma random variables

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 $Y$ {\it each} follow a Gamma law. This is related to Cram'er’s theorem which states that if $X$ and $Y$ are independent then $Z=X+Y$ follows a Gaussian law if and only if $X$ {\it and} $Y$ follow a Gaussian law. We prove that Cram'er’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.