Cram\'er theorem for Gamma random variables

TL;DR

This paper investigates whether the sum of two independent Gamma variables being Gamma implies each is Gamma, extending Cramér's theorem from Gaussian to Gamma distributions within Wiener chaos contexts.

Contribution

The paper proves Cramér's theorem for Gamma variables within fixed Wiener chaos, and provides an asymptotic version, highlighting limitations beyond this setting.

Findings

Cramér's theorem holds for Gamma variables in fixed Wiener chaos.

Counterexamples show the theorem does not hold in general for Gamma variables.

An asymptotic variant of the theorem is established.

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 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.