Cramér theorem for Gamma random variables

Abstract : 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 Cramér's theorem which states that if $X$ and $Y$ are independent then $Z=X+Y$ follows a Gaussian law if and only if $X$ {\it 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.
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Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2011, 16, pp.365-378
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Solesne Bourguin, Ciprian Tudor. Cramér theorem for Gamma random variables. Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2011, 16, pp.365-378. 〈hal-00605550〉

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